Advances in Numerical Modeling for Heat Transfer and Thermal Management: A Review of Computational Approaches and Environmental Impacts

Abstract

Advances in numerical modeling are essential for heat-transfer applications in electronics cooling, renewable energy, and sustainable construction. This review explores key methods like Computational Fluid Dynamics (CFD), the Finite Element Method (FEM), the Finite Volume Method (FVM), and multiphysics modeling, alongside emerging strategies such as Adaptive Mesh Refinement (AMR), machine learning (ML), reduced-order modeling (ROM), and high-performance computing (HPC). While these techniques improve accuracy and efficiency, they also increase computational energy demands, contributing to a growing carbon footprint and sustainability concerns. Sustainable computing practices, including energy-efficient algorithms and renewable-powered data centers, offer potential solutions. Additionally, the increasing energy consumption in numerical modeling highlights the need for optimization strategies to mitigate environmental impact. Future directions point to quantum computing, adaptive models, and green computing as pathways to sustainable thermal management modeling. This study systematically reviews the latest advancements in numerical heat-transfer modeling and, for the first time, provides an in-depth exploration of the roles of computational energy optimization and green computing in thermal management. This review outlines a roadmap for efficient, environmentally responsible heat-transfer models to meet evolving demands.

1. Introduction

Effective heat transfer and thermal management are essential across various industries, from electronics and automotive to renewable energy and building systems. As modern technology evolves, so does the need for efficient thermal control to prevent overheating, ensure safety, and maximize energy efficiency. The field of heat-transfer modeling has responded to these demands by adopting and advancing sophisticated computational techniques, making numerical modeling a cornerstone for understanding and controlling thermal behaviors in complex systems. This review focuses on the latest developments in numerical modeling for heat transfer, with particular attention to computational techniques and their environmental implications.

Traditionally, thermal transfer processes are delineated by three fundamental mechanisms: conduction, convection, and radiation [1]. Each mechanism performs an essential function depending on the material characteristics, the configurations of the system, and the operational conditions. However, the solutions to practical thermal transfer challenges are often complicated due to the interrelations of numerous factors, which encompass intricate geometries, transient boundary conditions, and coupled physical phenomena [2]. Numerical modeling has empowered researchers to address these complexities with increased precision, adaptability, and computational efficiency. Principal numerical methodologies, such as Finite-Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), Computational Fluid Dynamics (CFD), Lattice Boltzmann Method (LBM), and Smoothed-Particle Hydrodynamics (SPH) have become extensively employed in the discipline due to their capacity to effectively simulate fluid dynamics and thermal interactions. These methodologies, combined with multiphysics modeling, allow researchers to incorporate interactions among fluid, thermal, and structural phenomena, thereby enhancing the predictive capability of simulations.

Recent advances in computational methodologies have further augmented the capabilities of thermal management modeling [3]. AMR techniques, for example, have improved the simulation precision and efficiency of the simulation by refining the density of the mesh in regions characterized by elevated thermal gradients [4]. Furthermore, machine learning (ML) and artificial intelligence (AI) are making substantial contributions by providing data-driven methodologies that optimize computational resources, reduce simulation durations, and facilitate predictive maintenance [5]. Reduced-order modeling (ROM) has also attracted attention, as it simplifies intricate models while preserving fundamental dynamics, making real-time applications more practicable [6]. Ultimately, the function of high-performance computing (HPC) has been transformative, allowing extensive simulations that were previously unattainable due to computational constraints [7]. Specifically, constraints refer to the significant computational resources required, including high memory demands, processing power, and execution time, which previously limited the feasibility of extensive simulations. The implementation of high-performance computing (HPC) has helped to overcome these barriers, enabling more complex simulations.

Despite these advances, the environmental impact of computational modeling for heat transfer has raised growing concerns. High-fidelity modeling techniques frequently require considerable computational resources, which can result in significant energy expenditure and an increased carbon footprint [8]. The rapid expansion of AI-driven computations and data centers exacerbates these concerns, as highlighted in the International Energy Agency (IEA) report “Energia 2024”, which predicts that global energy demand for AI and cryptocurrency could double by 2026, equaling the electricity consumption of Japan. Major tech companies—Microsoft, Meta, Google, and Amazon—are rapidly expanding data center infrastructure, contributing to unprecedented power demands. A 2024 Goldman Sachs report forecasts that U.S. data centers will consume 8% of the country’s electricity by 2030, up from 3% in 2022, straining existing power grids. This surge in energy demand has driven corporations toward nuclear power, with Amazon acquiring a nuclear-powered data center in Pennsylvania, and Microsoft securing a 20-year nuclear energy deal. However, concerns over grid stability have led to regulatory pushback, such as the U.S. Federal Energy Regulatory Commission (FERC) rejecting a proposal for Amazon to source energy from a nuclear plant. These developments emphasize the urgent need for sustainable computing solutions, including energy-efficient algorithms, low-power hardware, and the integration of renewable energy sources in data centers, to mitigate the growing environmental impact of high-performance simulations. This review examines not only the technical advancements in numerical modeling but also the sustainability issues that accompany them. Efforts to mitigate ecological consequences, such as energy-conserving algorithms, low-power hardware, and the implementation of renewable energy-powered data centers, are progressively significant. These practices, known as sustainable computing, aim to balance computational accuracy with environmental responsibility, helping to mitigate the carbon footprint associated with high-performance simulations.

In addition to sustainability concerns, several challenges remain in the field. Balancing the compromises between precision and computational expense is a persistent concern, particularly in simulations that require elevated spatial and temporal resolutions. Capturing real-world conditions accurately in models—such as transient heat flows, variable boundary conditions, and multi-physics interactions—remains a complex task. Furthermore, the integration of AI and ML into heat-transfer modeling presents its own set of challenges, particularly with respect to data quality and availability. High-quality data are essential for training robust machine learning models, but obtaining accurate and diverse datasets can be difficult in many engineering applications [9]. These include turbulent convection, phase-change processes such as boiling and condensation, and radiative heat transfer in participating media. In these cases, experimental measurements are often limited by sensor accuracy, spatial and temporal resolution constraints, and complex boundary conditions.

Looking ahead, the promising future directions in numerical modeling for heat transfer include quantum computing, which has the potential to revolutionize thermal simulations by offering computational power beyond traditional HPC systems. While quantum computers are expected to provide advantages only for certain types of computationally intensive problems, some heat-transfer phenomena, such as nanoscale thermal transport, fall into this category. Specifically, solving the Boltzmann transport equation (BTE) for phonon transport in semiconductors or addressing quantum many-body interactions in thermal systems are areas where quantum computing could be beneficial. These applications involve complex, high-dimensional calculations that are challenging for classical methods, making them potential candidates for quantum acceleration [10]. Quantum algorithms could provide faster solutions to complex heat-transfer problems, opening new possibilities for adaptive models functioning in real time that respond to dynamic environmental conditions [11]. Green computing initiatives are also expected to play an important role, with advances in low-energy processors, optimized algorithms, and collaborative open-source platforms that foster environmentally responsible practices [12]. These developments signify a shift toward a more sustainable approach to computational modeling, one that seeks to minimize environmental impact, while advancing technical capabilities.

This review presents a thorough examination of the current progress in numerical modeling methodologies pertaining to heat transfer and thermal management. It discusses their applications, emerging technologies, environmental challenges, and potential future directions, offering researchers and engineers insights into the development of efficient and sustainable thermal management solutions.

2. Overview of Numerical Modeling Techniques in Heat Transfer

Numerical modeling methodologies in the domain of heat transfer are essential for the precise forecasting and regulation of thermal phenomena across a multitude of applications. These methodologies inspire engineers and scholars to replicate intricate thermal scenarios, refine designs, and increase system efficiency. Among the foremost techniques, FDM systematically discretizes heat-transfer equations over a well-structured grid, making it particularly effective for conduction-dominant scenarios within simplistic, uniform geometries [13]. However, the dependency of FDM on structured grids limits its flexibility in handling highly irregular geometries and complex boundary problems, making it less suitable for such scenarios, compared to other methods such as FEM. FEM mitigates this constraint by partitioning the domain into smaller interconnected elements, thus providing enhanced adaptability for intricate geometries and transient heat phenomena [14]. FEM is widely utilized in sectors such as automotive [15] and aerospace [16] engineering, in which precise modeling of unconventional structures and boundary conditions is of critical importance.

For scenarios that encompass substantial fluid–thermal interactions, CFD proves to be invaluable, as it effectively simulates convective heat transfer and the interactions that occur between fluids and surfaces. CFD is routinely employed in the domains of electronics cooling, HVAC systems, and energy systems, in which it facilitates the analysis and optimization of fluid–thermal behavior to enhance heat dissipation and energy efficiency. In recent years, innovative techniques have further propelled advancements within the realm of heat-transfer modeling. AMR, for instance, dynamically modifies grid resolution to effectively capture regions characterized by high thermal gradients, thereby improving model precision without incurring excessive computational burdens. Moreover, ML methodologies have introduced data-driven models that facilitate rapid predictions for complex thermal systems, significantly decreasing computational time while maintaining acceptable accuracy levels. Additionally, ROM simplifies comprehensive simulations into lower-dimensional constructs, allowing for faster analyses without sacrificing critical dynamic characteristics.

Each of these methodologies has distinct advantages and limitations, and their applications are selected based on system complexity, precision requirements, and the computational resources at hand. The selection of the most suitable technique is highly dependent on the specific challenges presented by the thermal management issue and the performance expectations related to the application in question. To provide a clearer understanding of the strengths and limitations of the methods presented in the subsequent parts of Section 2,

Table 1. Structured comparison of the methods.

Method	Scope of Application	Computational Cost	Accuracy	Advantages	Disadvantages
FDM	Heat conduction, simple geometries	Low to moderate	Moderate	Easy to implement, sufficiently efficient for structured grids	Limited to structured grids, less flexible for complex geometries
FEM	Structural-thermal interactions, complex geometries	Moderate to high	High	High accuracy, sufficiently flexible for complex geometries	Computationally expensive, requires meshing
FVM	Fluid and thermal simulations, CFD applications	Moderate	High	Conservative properties, suitable for control volume approach	Requires complex discretization for accuracy
CFD	General fluid flow and thermal interactions	Moderate to high	High	Sufficiently versatile for fluid dynamics problems	Computationally intensive for transient simulations
LBM	Micro-scale and porous media flow, multiphysics applications	High	Moderate to high	Suitable for parallel computing, captures complex boundaries	Requires large computational resources, limited industrial adoption
SPH	Free-surface flows, highly deformable materials	Moderate to high	Moderate	Mesh-free, sufficiently efficient for complex interfaces	High computational cost, lower accuracy in some cases

has been added to summarize their key characteristics.

2.1. Finite-Difference Method (FDM)

The FDM constitutes a numerical approach extensively employed for the resolution of heat conduction phenomena, primarily owing to its inherent simplicity and efficacy in accommodating various boundary conditions and structured geometrical configurations. This technique involves discretization of the heat equation across a grid in which derivatives are approximated using finite differences, thereby facilitating the numerical resolution of intricate conduction issues. FDM proves particularly beneficial in contexts in which analytical solutions are challenging to derive or are altogether nonexistent. In the subsequent discussion, the implementation and merits of FDM in modeling conduction phenomena are elaborated on comprehensively.

FDM is a numerical approach to solving partial differential equations (PDEs) by approximating derivatives with finite differences. The process begins with discretization of the domain, dividing the problem into a grid of discrete points in 1D, 2D, or 3D. Derivatives in the governing equations are replaced by finite-difference approximations obtained from Taylor series expansions, such as forward, backward, or central difference schemes. These approximations are then substituted into the PDE to form a discrete system of algebraic equations. This system is formulated as Au = b, where u is the vector of unknowns, A is the coefficient matrix, and b contains the source terms and boundary conditions. Boundary conditions are applied by modifying the equations at the boundaries. Finally, numerical techniques such as Gaussian elimination, LU decomposition, or iterative solvers (e.g., Jacobi or Gauss–Seidel) are applied to solve the system for the unknowns. The solution is then analyzed and refined by adjusting the grid space or time steps, and the convergence and precision are then verified [17,18]. The flowchart of the steps in the FDM is presented in Figure 1. For a detailed introduction to finite-difference methods in heat transfer, readers may refer to Ozisik et al. [19].

FDM is used in solving inverse boundary value problems in 2D steady-state anisotropic heat conduction. It helps reconstruct missing boundary conditions and thermal fields from over-prescribed data, demonstrating its utility in complex boundary scenarios [20]. The FDM is proficiently utilized in addressing two-dimensional heat conduction challenges, yielding commendable outcomes across a spectrum of boundary conditions and geometries. The methodology is executed through algorithms, such as those devised in Matlab (version 2017b), to tackle issues related to temperature distributions in plates with and without heat generation. The numerical solutions derived through FDM have exhibited remarkable precision, with the relative errors recorded being as minimal as 0.003 or 0.03 for scenarios without or including heat generation, respectively, in the steady-state case [21]. The FDM is further adapted to encompass transient heat conduction issues, wherein time-dependent fluctuations in temperature are taken into account. An innovative space–time generalized FDM incorporates direct space–time discretization, facilitating the investigation of transient heat conduction in both homogeneous and heterogeneous materials. This methodology has evidenced superior accuracy and efficiency, as corroborated through comparative analyses with both analytical and alternative numerical outcomes [22]. The FDM is employed to address hyperbolic heat conduction equations which consider the finite velocity of heat propagation. Both explicit and implicit FDM calculation schemes are used, the technique demonstrating its efficacy in approximating time derivatives and managing the boundary and initial conditions [23].

The FDM is characterized by its straightforward implementation and adaptability to a diverse range of boundary conditions, including Neumann, Dirichlet, and mixed conditions. This inherent flexibility makes it applicable to a wide spectrum of heat conduction challenges [24]. The methodology shows proficiency in addressing irregular geometrical configurations, particularly when combined with meshless techniques. For example, a meshless variant of FDM has been developed for conjugate heat-transfer challenges, facilitating the resolution of heat conduction in complex geometries without the need for structured grids [24]. FDM can be customized to simulate heat conduction in materials characterized by varying properties, such as functionally graded materials (FGMs). This adaptation capacity is essential for the accurate representation of real materials that possess spatially heterogeneous thermal properties [25,26]. FDM has been compared with the Smoothed Particle Hydrodynamics (SPH) method, which is another numerical approach used to solve heat conduction problems. Although SPH offers advantages in certain scenarios, FDM remains a robust choice due to its established framework and ease of use. Although previous studies have explored comparisons between these methods, including the analysis in [27], a more comprehensive evaluation considering computational cost and accuracy is necessary for a rigorous assessment.

Recent innovations in FDM include the development of hybrid methods that integrate FDM with other numerical techniques to enhance performance. For example, the hybrid Restarting FDM–SVR–Lanczos method combines FDM with machine learning to solve heat flow problems in welding more effectively [28].

Figure 2 presents a summary of the individual topics discussed in relation to FDM.

Although FDM represents a formidable instrument for modeling conduction phenomena, it is not devoid of limitations. The method may experience numerical instability and inaccuracies, particularly in scenarios that involve sharp gradients or discontinuities. However, recent advances, such as unconditionally stable explicit FD methods [29], have addressed this limitation. These methods maintain second-order accuracy even when applied to heterogeneous media, enhancing the reliability of FDM in complex heat-transfer problems. Furthermore, recent studies [30,31] have extended and applied FD-type methods in various engineering applications, demonstrating their practical utility. Despite these developments, the precision of FDM solutions is intricately linked to grid resolution, potentially resulting in increased computational expenditures for fine grids. Notwithstanding these challenges, FDM remains an invaluable methodology for addressing a wide array of heat conduction issues, especially when integrated with contemporary computational techniques; for example, it is widely used in LBM.

2.2. Finite Element Method (FEM)

FEM represents a sophisticated numerical approach that is used to address the intricate problems associated with heat transfer, particularly in scenarios in which analytical solutions are impractical. FEM is of significant importance within the domain of heat transfer as a result of its proficiency in managing complex geometries, boundary conditions, and variations in material properties. This methodology involves the discretization of a domain into smaller, manageable elements, thereby facilitating the approximation of solutions to the PDEs that govern heat-transfer phenomena. The adaptability and accuracy of FEM renders it essential in both engineering and scientific endeavors related to heat transfer.

FEM is a numerical technique for solving PDEs by dividing the domain into smaller, simpler parts, called elements, and defined by the nodes. The process begins by discretizing the domain; in this, the problem is divided into a finite number of elements that collectively form a mesh. An appropriate element type (e.g., linear, quadratic) and corresponding shape functions are selected to approximate the distribution of unknown variables within each element on the basis of values on the nodes. The governing PDE is then rewritten in its weak (variational) form by integrating against a test function, ensuring that the solution satisfies the equation in an average sense. For each element, the element stiffness matrix and the load vector are computed using the shape functions and the weak form of the PDE, and these contributions are assembled into a global system of algebraic equations (a stiffness matrix is the left part of the system K, and the load vector f is the right part). Boundary conditions (e.g., Dirichlet or Neumann) are incorporated to modify the global system, resulting in a linear system of equations, Ku = f, where u represents unknown nodal values. The system is solved using numerical techniques, and the solution is post-processed to compute derived quantities (e.g., temperature or heat fluxes) and to visualize or interpret the results [32,33]. The individual stages of the FEM modeling process are presented in Figure 3.

FEM is particularly useful for solving heat conduction problems in complex 3D geometries, in which traditional analytical methods fall short. For example, it can model intricate shapes such as L-shaped blocks, effectively capturing localized temperature gradients and discontinuities [34].

In the context of natural convection, FEM has been used to study the heat transfer between a cold circular cylinder and a heated corrugated cylinder, demonstrating sensitivity to parameters such as Rayleigh number and cylinder inclination [35].

FEM excels at addressing heat transfer in irregularly shaped domains, which are common in real-world applications. By partitioning these domains into smaller elements, FEM can systematically solve heat-transfer equations, as demonstrated in studies involving irregular 2D geometries [13].

The method’s ability to model non-stationary heat conduction in composite materials further highlights its adaptability to complex material properties and geometries [36].

The application of open-source computational tools in FEM facilitates the effective modeling of thermal transfer phenomena within intricate systems, exemplified by heat sinks utilized in the electronics sector. This methodological framework capitalizes on advanced computing capabilities to address the challenges associated with conduction, convection, and radiation, providing a financially viable substitute for proprietary software solutions [37].

FEM can be integrated with the Boundary Element Method (BEM) to enhance the examination of structures across multiple scales. This integration improves the precision of solutions and reduces computational expenses, establishing it as a formidable option for complicated heat-transfer challenges [38].

FEM is applied in various fields, including bio-heat transfer for medical treatments such as Magnetic Fluid Hyperthermia, in which it helps to target cancerous tissues without harming healthy ones [39].

In the automotive industry, automated meshing techniques have been developed to handle complex geometrical configurations efficiently. This automation is crucial to keeping up with the fast-paced development cycles in motorsports and other automotive applications [40]. Transient thermal simulations are vital to managing the thermal behavior of the automotive underhood and underbody components. FEM helps to accurately model the thermal mass and interactions between different materials, which is essential for vehicle development [41].

FEM is widely used for transient thermal analysis in aerospace, such as in the design of permanent magnet motors and induction motors. These analyses are critical to the optimization of motor performance under varying load conditions and ambient temperatures [42,43].

Inverse heat conduction analysis using FEM is applied to atmospheric reentry vehicles to predict aerodynamic heating and ensure structural integrity under extreme conditions [44]. The inverse method has been used to identify the heat-transfer coefficient under different conditions of water spray pressure, water flux, and nozzle-to-surface distance [45]. Additionally, the solution strategy for the inverse determination of the specially-varying heat-transfer coefficient on the water-cooled plate has been developed [46]. Inverse solutions to vertical plate cooling in air were applied in a comparative study [47].

Figure 4 presents a summary of the individual topics discussed in relation to FEM.

Although FEM serves as a robust instrument for the analysis of heat-transfer phenomena, it is not devoid of inherent constraints. The precision of FEM-derived solutions is profoundly influenced by the quality of the mesh configuration and the computational resources at one’s disposal. In certain instances, the methodology may require substantial computational capacity, particularly when dealing with highly detailed or intricate models. Furthermore, the establishment and interpretation of FEM frameworks can be complex and require a high level of expertise in both the technique and the specific area of application. Notwithstanding these obstacles, FEM continues to be a fundamental element in the analysis and simulation of heat-transfer challenges across diverse industries and scholarly fields. FEM requires large, or even huge, computational resources and simulations can take a long time. Often, results of FEM calculation serve as databases for AI, including ANN, ML, and DL, and further application of trained ANN as ROM.

2.3. Computational Fluid Dynamics (CFD)

CFD is integral to the regulation of convective heat transfer, fluid–thermal interactions, and intricate geometrical configurations in a range of applications. Its importance is particularly pronounced in domains such as electronic cooling, battery thermal management, and HVAC (heating, ventilation, and air conditioning) systems, in which meticulous thermal regulation is vital for optimal performance and safety. CFD facilitates comprehensive simulations that can forecast thermal behavior, improve design efficiency, and improve overall system efficacy. A detailed examination of CFD applications in these specific areas is provided in the following.

CFD is a numerical approach to the analysis and prediction of fluid behavior; it solves governing equations of fluid motion, such as the Navier–Stokes equations. The process begins with problem definition, which involves specifying the geometry, fluid properties, boundary conditions (e.g., inlets, outlets, walls) and initial conditions for time-dependent problems. The domain is then discretized into a mesh of control volumes or elements, ensuring adequate refinement in critical regions such as boundaries or areas with steep gradients. The governing equations, including the continuity, momentum, and energy equations, are defined, together with any necessary models, such as turbulence or multiphase flow models. These equations are discretized using numerical methods such as FVM, which convert partial differential equations into algebraic equations at each point in the grid. The resulting equations are solved using iterative techniques (e.g., explicit or implicit methods), ensuring convergence and proper coupling between pressure and velocity fields via algorithms like SIMPLE or PISO. The results are then post-processed using visualization tools to analyze velocity fields, pressure distributions, and derived quantities like drag or heat-transfer rates. Finally, iterative refinement of the mesh, boundary conditions, or models is performed to improve accuracy, repeating the process until the desired solution is achieved [48,49]. The stages of the CFD modeling process are presented in Figure 5.

Turbulence modeling is a fundamental aspect of CFD simulations, particularly in engineering applications in which fluid flow exhibits chaotic and unsteady behavior. Accurate prediction of turbulence is essential for applications such as aerodynamics, HVAC systems, and thermal management in electronics. Various turbulence models are used in CFD to address different flow regimes, including Reynolds-Averaged Navier–Stokes (RANS) models, Large Eddy Simulation (LES), and Direct Numerical Simulation (DNS) [50,51]. RANS models, such as the k-ε and k-ω models, offer a balance between computational cost and accuracy, which make them widely used in industrial applications such as HVAC systems and vehicle aerodynamics [52,53]. LES provides greater accuracy by resolving large-scale turbulent structures while modeling smaller eddies, making it suitable for indoor environment control and aerospace applications [54]. On the contrary, DNS offers the highest fidelity by directly solving all turbulent scales, but it comes at a significantly higher computational cost, making it practical only for small-scale turbulence studies [55].

Additionally, multi-physics coupling plays a vital role in numerous real-world applications, where CFD must be integrated with other physical models. Fluid–solid coupling is critical in structural integrity analysis, in which fluid forces interact with deformable structures, leading to applications in aerospace, biomedical devices, and civil engineering [56,57].

Multiphase flow modeling is another key aspect; in this application, CFD is utilized to simulate interactions between different fluid phases, such as liquid–gas and liquid–solid flows. This is particularly important in processes such as fuel injection, boiling, and slurry transport [58,59]. CFD models such as Volume of Fluid (VOF) [60], Eulerian–Eulerian [61], and Eulerian–Lagrangian [62] approaches are commonly employed to handle these complex interactions by tracking phase boundaries and interfacial dynamics.

Furthermore, CFD is extensively used in modeling chemically reactive flows, such as combustion processes and electrochemical reactions in batteries and fuel cells, in which the couplings between fluid flow, chemical kinetics, and heat transfer are crucial to accurate predictions [63]. The Eulerian–Eulerian two-phase flow model has been widely adopted to model solid oxide fuel cells and proton exchange membrane fuel cells (PEMFCs), allowing for detailed resolution of electrochemical reaction zones and gas-phase transport phenomena [64].

CFD is used to analyze cold plate designs for electronics cooling, particularly those produced by electrochemical additive manufacturing (ECAM). These designs offer improved thermal performance compared to traditional methods by enabling complex geometries that improve heat dissipation [65]. CFD and conjugate heat-transfer analyses are used to evaluate different fin geometries in heat sinks. This helps identify configurations that minimize thermal resistance and pressure drop, such as slanted mirror geometry, which provides superior performance [66]. The role of thermal design with CFD in electronics has evolved significantly since the 1980s; it has become a critical tool for predicting operating temperatures and refining thermal designs during product development [67].

CFD simulations are used to compare different battery thermal management systems (BTMS), such as indirect liquid-based cooling and direct immersion cooling. These simulations help assess thermal gradients and identify hotspots, which are crucial for maintaining battery safety and performance [68]. The immersion coupled direct cooling (ICDC) method, analyzed through CFD, shows enhanced cooling performance by extending the optimal working duration of batteries. This method reduces battery temperature more effectively than natural convection or traditional immersion cooling [69]. CFD is utilized to design and optimize mini-channel liquid cooling plates with disturbance structures. These designs improve heat dissipation and flow field performance, ensuring uniform temperature distribution throughout the battery module [70].

CFD can simulate airflow and heat transfer in complex duct systems, optimize the placement of vents and diffusers, and improve energy efficiency by predicting thermal loads and system responses. The food industry uses CFD to optimize processes such as bread baking, cooling, and spray drying. This improves the quality and safety of food products while reducing production costs [71].

Recent advances in CFD include the integration of thermal energy systems with renewable energy sources, leading to the development of hybrid systems [72]. Future research in CFD aims to develop materials with higher thermal storage capacity and enhance heat-transfer rates, further optimizing system designs [72]. Additionally, the continuous development of advanced numerical models will further enhance the accuracy and applicability of CFD simulations in thermal management and other fields [73].

Figure 6 presents a summary of the individual topics discussed in relation to the CFD method.

2.4. Finite Volume Method (FVM)

FVM constitutes a computational approach extensively employed for the resolution of partial differential equations, especially within the realms of thermal transfer and fluid dynamics. This methodology is predicated upon the conservation principle, whereby the spatial domain is partitioned into a finite set of control volumes and the governing equations are systematically integrated over these defined volumes. The efficacy of this method is particularly pronounced in its ability to adeptly manage intricate geometries while simultaneously ensuring the conservation of fluxes at the interfaces of control volumes. The application of FVM spans a diverse array of heat-transfer contexts, encompassing acoustic enclosures, cold plates, and porous media, thereby illustrating its remarkable versatility and operational effectiveness.

FVM divides the domain into discrete “control” volumes. The integral form of the conservation equations is applied to each control volume, ensuring that the flux entering a volume equals the flux leaving it, taking into account any sources or sinks within the volume [74]. The method involves discretizing the spatial domain into a mesh, in which the governing equations are solved. This can be accomplished using structured or unstructured meshes, allowing for flexibility in the handling of complex geometries [75]. FVM inherently conserves mass, momentum, and energy, making it particularly suitable for heat-transfer problems in which these principles are critical [76]. The flowchart of the FVM modeling process is presented in Figure 7.

FVM has been used to model heat transfer in acoustic enclosures, helping to predict temperature distributions and optimize cooling strategies to prevent the overheating of machinery [77]. In the automotive industry, FVM is applied to enhance heat dissipation on cold plates for lithium batteries. By optimizing channel designs, such as by using wavy channels, FVM helps improve thermal performance significantly [78]. The FVM is also employed to solve radiative heat-transfer problems, particularly in media that emit, absorb, and scatter radiation. This application is crucial in systems with complex boundary conditions and varying material properties [79]. The method is effective in predicting transport phenomena in porous media, including heat and mass transfer coupled with chemical and biological reactions. This is particularly useful in environmental and energy applications, such as composting and combustion processes [80].

Figure 8 presents a summary of the individual topics discussed in relation to FVM.

Although FVM is exceedingly effective in guaranteeing conservation and managing intricate geometries, it may necessitate substantial computational resources, particularly for three-dimensional and transient issues. Furthermore, the precision of FVM can be affected by the caliber of the mesh and the numerical methodologies used for discretization [74,76]. Notwithstanding these obstacles, FVM continues to be a favored selection in numerous engineering applications owing to its resilience and flexibility.

2.5. Lattice Boltzmann Method (LBM)

LBM is a computational technique that has acquired significance for modeling fluid dynamics and thermal transfer phenomena. Deriving from the lattice gas automata methodology, LBM employs a mesoscopic framework to represent the distribution of particle velocities across a lattice framework, thereby enabling the recovery of macroscopic transport equations. Its pertinence to thermal transfer is emphasized by its capacity to manage intricate geometries, boundary conditions, and multi-physical field interactions with relative simplicity, in comparison to conventional computational fluid dynamics techniques. This makes LBM especially apt for simulating a variety of thermal transfer phenomena, encompassing conduction, convection, and radiation, in heterogeneous media and configurations. It also allows for simulations of complex problems in conjunction with chemical reactions, diffusion, etc.

The same LBM algorithm is applied to all cases of modeled problems, following a cyclic process (Figure 9) that includes the following steps:

• Calculation of macroscopic variables such as density (ρ), velocity (u), temperature (T), concentration (C), and others.
• Evaluation of the equilibrium distribution function f^eq for the modeled variables.
• Collision operation, which involves the determination of the output distribution function f^out.
• Streaming operation, which transfers the distribution function to the appropriate cells, sites, or nodes.
• Application of boundary conditions.

The cycle typically begins with the calculation of macroscopic variables and equilibrium distribution functions, as well as the assignment of output functions, and can technically start at any stage.

The method is fundamentally based on the discretization of space and time. A regular grid is applied to the domain, in which the grid is square in two dimensions or cubic in three dimensions, with the spacing between adjacent nodes set to one. Similarly, the time step length is also set to one. A crucial component of the system is the choice of velocity model. In one-dimensional space, models such as D1Q2 and D1Q3 are commonly used; in two-dimensional space, D2Q4, D2Q5, and D2Q9 are employed. For three-dimensional problems, models such as D3Q6, D3Q7, D3Q15, D3Q19, and others are utilized. Here, “D” represents the dimensionality, while “Q” denotes the number of velocity directions. Additional details about LBM can be found in [81].

LBM functions on a mesoscopic scale, facilitating the connections between microscopic particle dynamics and macroscopic fluid phenomena. This enables the proficient simulation of intricate fluid flows and thermal transfer mechanisms without the need to directly solve the Navier–Stokes equations [82]. LBM is adept at managing complex geometries and boundary conditions, which is crucial to accurately modeling heat transfer in irregular domains such as porous media and curved boundaries [82,83]. LBM has been effectively applied to simulate conjugate heat transfer, in which heat conduction in solids interacts with fluid flow and heat transfer in adjacent fluids. This is particularly useful in engineering applications that involve heat exchangers and thermal management systems [84]. The method has been extended to handle coupled radiation-conduction heat-transfer problems, especially in media with complex boundaries and graded-index materials. This capability is crucial for applications in thermal management and energy systems in which radiative heat transfer plays an important role [83,85]. LBM has been adapted to simulate phase-change phenomena, such as boiling, which involve complex interactions between heat transfer and fluid dynamics. This is achieved through specialized models that account for the effects of thermodynamic and surface tension on phase-change processes [86]. This method was also applied to the modeling of heat transfer during the phase transformations that occur in materials [87,88].

LBM provides high accuracy in predicting temperature and heat-flux distributions, often outperforming traditional methods such as FVM in terms of computational speed and ease of implementation [89]. The adaptability of the approach facilitates the integration of various physical phenomena, such as thermal emission and natural convection, within a cohesive framework. This makes it a multifaceted instrument usable for modeling an extensive array of heat-transfer situations [90].

The current status of LBM applied to aerodynamic, aeroacoustic, and thermal flows was presented in [91]. The modeling of natural convection heat transfer on a large scale became possible due to new calculation schemes applied in thermal LBM [92].

Figure 10 presents a summary of the individual topics discussed in relation to LBM.

Although LBM presents numerous benefits, it is not devoid of constraints. The precision of the methodology can be susceptible to the selection of relaxation durations and mesh granularity, particularly in simulations encompassing substantial temperature differentials or intricate phase-transition phenomena. Moreover, although LBM is computationally proficient, it may require considerable memory resources for high-resolution simulations, especially in three-dimensional contexts. Notwithstanding these obstacles, ongoing research and advancement continues to expand the capabilities and applicability of LBM in thermal transfer and fluid dynamics. A very important feature of the method is its adaptability to parallelization, which opens wide possibilities for fast and effective calculations.

2.6. Smoothed-Particle Hydrodynamics (SPH)

Smoothed Particle Hydrodynamics (SPH) is a mesh-free, Lagrangian method for solving fluid dynamics and heat-transfer problems by representing a continuum as discrete particles. Each particle carries properties such as mass, velocity, and temperature, interacting with its neighbors via a smoothing kernel function. The process begins with particle initialization, assigning properties and discretizing the domain. The kernel approximation estimates field variables, while density computation ensures mass conservation. Forces are evaluated by solving the Navier–Stokes equations, incorporating pressure, viscosity, and external forces. If heat transfer is modeled, the energy equation updates the temperatures. The system evolves through time integration using explicit schemes like Euler or Verlet. Boundary conditions are enforced via ghost particles or repulsive forces, and particle properties are iteratively updated until the simulation reaches convergence [93].

The flowchart of the steps in the SPH is presented in Figure 11.

This method has been applied across various fields, including Computational Fluid Dynamics, astrophysics, and engineering, demonstrating its adaptability and efficiency in handling diverse thermo-fluid challenges.

The SPH meshless approach allows it to handle complex geometries and free surface flows effectively, making it suitable for applications such as fluid–structure interactions and multiphase flows [94,95]. The Lagrangian framework of SPH enables a straightforward handling of advection and complex boundary conditions, which is beneficial in simulating multiphase flows and fluid–solid interactions [94,96]. SPH has been successfully applied in various fields, from tunnel engineering to astrophysics, demonstrating its ability to model a wide range of fluid dynamics scenarios [95].

Almério et al. provided a detailed explanation of the implementation of the SPH method to solve the heat equation in an unsteady state [97]. In addition, they compared the SPH with the FDM in terms of accuracy and computational cost. Similarly, K.C. Ng et al. applied the SPH method to estimate heat transfer in complex geometries [98]. The SPH method has also been utilized in various engineering applications, such as additive manufacturing [99].

The REMIX SPH scheme addresses problems related to mixing and instability growth in density discontinuities, improving the treatment of interfaces between dissimilar materials, and capturing hydrodynamic behaviors with greater precision [100]. Incorporating entropy into SPH allows for pressure dependence on temperature, enhancing the method’s capability to simulate thermo-fluid dynamics with energy conservation and entropy growth considerations [101].

SPH simulations can suffer from particle clustering due to tensile instabilities, which can affect the accuracy and stability of long-term predictions. Recent advancements in neural SPH models aim to mitigate these issues by integrating machine learning techniques [102]. Although SPH is computationally intensive, advances in parallel computing, such as the use of compute shaders, have improved its performance, making real-time simulations more feasible [103].

SPH continues to evolve; ongoing research addresses its limitations and expands its applicability. While it offers significant advantages in handling complex thermo-fluid scenarios, challenges such as computational efficiency and stability remain areas of active development. The integration of machine learning and hybrid numerical methods presents promising avenues for enhancing SPH’s capabilities, potentially leading to more accurate and efficient simulations in the future.

Figure 12 presents a summary of the individual topics discussed in relation to the SPH method.

3. Recent Advances in Computational Approaches for Heat Transfer

3.1. Adaptive Mesh Refinement (AMR)

AMR is a computational methodology employed to increase the precision and efficacy of numerical simulations in thermal transfer and fluid mechanics. AMR dynamically modifies the resolution of the computational lattice in accordance with the solution’s exigencies, permitting finer lattices in regions characterized by steep gradients or intricate features, while coarser lattices are utilized in other areas. This versatility engenders substantial advances in computational efficacy and precision, making AMR an invaluable instrument in the domain of computational thermal transfer.

The AMR process begins by initializing a base grid, where the problem domain is defined and discretized into a coarse and uniform grid with initial values for key variables such as velocity, pressure, or temperature [104]. An initial computation is performed to solve the governing equations, identifying regions requiring higher resolution, such as areas with steep gradients or shock waves [105]. Based on error estimation and refinement criteria, such as gradient-based or residual-based methods, regions exceeding a defined threshold are refined by the subdivision of cells into smaller cells while ensuring smooth transitions between coarse and fine regions [106]. The governing equations are then recomputed on the refined grid, using interpolation or reconstruction to transfer information between the grids [107]. This process of iterative refinement and coarsening continues, dynamically refining regions with persistent errors and coarsening those in which high resolution is no longer needed, optimizing computational efficiency [108]. Once the simulation reaches the desired accuracy, post-processing is performed to analyze and visualize the results, highlighting the refined regions and the behavior of the solution, the refinement being repeated as necessary to meet the convergence and accuracy requirements [104]. The flowchart of the AMR is presented in Figure 13.

AMR allows for higher resolution in areas in which the solution exhibits rapid changes, such as boundary layers or shock fronts, leading to more accurate results without the need for a uniformly fine grid throughout the domain [109,110]. By concentrating computational resources only where needed, AMR reduces the total number of points in the grid and the computational cost, making it feasible to solve large-scale problems with limited computational resources [111,112]. AMR can dynamically adjust the grid during simulation, responding to evolving features in the flow field or thermal field, which is particularly useful in transient simulations in which regions of interest can change over time [113].

AMR is utilized in electronic devices to effectively manage heat dissipation, which is crucial to maintaining device performance and longevity. The goal-oriented AMR algorithm proposed by Wang et al. [114] focuses on specific metrics rather than global error, allowing for targeted refinement in areas with high thermal gradients, such as surface mount packages. This approach eliminates unnecessary refinement, improving convergence behavior and computational efficiency. In coupled thermal–fluid systems, AMR is integrated with adjoint-based solvers to optimize heat transfer while controlling mechanical power dissipation. Gallorini et al. [115] demonstrate the use of AMR in a finite volume solver for aero-thermal optimization of cooling systems. The hierarchical nonconforming h-refinement strategy ensures mesh independence and conservation across topology changes, as validated through two- and three-dimensional test cases. Adaptive Mesh Refinement (AMR) is utilized in the computational modeling of industrial gas turbine combustors to effectively capture the intricate interactions between fluid dynamics and thermal transfer. McManus et al. implement AMR within the framework of Large Eddy Simulations (LES) to dynamically optimize the computational mesh in response to variations in mixture fraction and reaction progress. This methodology facilitates a double improvement in computational efficiency compared to static mesh configurations, while simultaneously preserving precision in the prediction of flow characteristics and flame temperatures [116].

Figure 14 presents a summary of the individual topics discussed in relation to FDM.

Although AMR offers significant advantages in handling high thermal gradients, it is essential to consider the computational overhead associated with dynamic mesh adjustments. The balance between precision and computational cost is a critical factor in the successful implementation of AMR. Additionally, the choice of refinement criteria and the ability to maintain conservation across mesh changes are crucial for achieving reliable results. As AMR techniques continue to evolve, they hold the potential to further enhance the efficiency and accuracy of simulations in various heat-transfer applications.

3.2. Artificial Intelligence and Machine Learning (AI/ML)

AI and ML have become essential in the progress of heat-transfer modeling, providing innovative solutions for predictive analytics and accelerated simulations, and reducing computational costs. These technologies use data-centric methodologies to improve the precision and efficacy of heat-transfer simulations, which are vital for optimizing thermal systems. The incorporation of AI and ML into heat-transfer modeling is revolutionizing conventional techniques by delivering faster and more reliable forecasts, thus facilitating improved design and operational determinations.

The sudden successes of ML and then deep learning (DL) from 2012 to 2015 were not due to some new discovery or breakthrough in theory (many people were describing deep neural networks and backpropagation as far back as the 1950s), but occurred for two reasons: the massive increase in computer power (including a 100x speed increase achieved by moving to GPUs) and the availability of massive amounts of training data. However, in engineering applications, obtaining such large, high-quality datasets is challenging. Experimental data are often limited by sensor inaccuracies, sparse measurements, and high costs, while numerical datasets require computationally expensive simulations that may not provide full-coverage data for complex scenarios. These limitations impact the generalization capabilities of AI/ML models, as insufficient or biased training data can lead to overfitting and reduced performance in real-world applications.

ML involves a systematic process used to develop models that make predictions or decisions based on data. The process starts by defining the problem, identifying the type of task (e.g., supervised learning for predictions, unsupervised learning for pattern discovery, or reinforcement learning for decision optimization), and establishing performance metrics such as accuracy or precision [117]. Next, data collection and preparation involve the gathering of raw data, cleaning it to address missing values or outliers, engineering relevant features, and splitting the data into training, validation, and test sets [118]. To address data limitations, techniques such as transfer learning can be employed, in which a pre-trained model from a related domain is fine-tuned with a limited amount of new data to enhance performance. This approach has been particularly useful in cases where the acquisition of large-scale experimental datasets is impractical. Additionally, domain adaptation and physics-informed machine learning methods help improve model generalization by incorporating physical constraints into AI predictions. Then, an appropriate algorithm or model is selected based on the problem, ranging from linear models and tree-based methods to neural networks or clustering algorithms [119]. The chosen model is trained on the training data by optimizing its parameters to minimize a loss function. During validation and tuning, the model is evaluated on validation data, hyperparameters are optimized, and techniques such as cross-validation or regularization are used to prevent overfitting [120]. The final model is tested on the test dataset to ensure that it generalizes well to unseen data. Once validated, the model is deployed in production systems (e.g., APIs or embedded devices) and is monitored in real time for performance [121]. Finally, monitoring and updating ensure that the model remains accurate over time, with periodic retraining on new data to handle changes or mitigate performance degradation. Individual steps in AI/ML are presented in Figure 15.

ML frameworks, including decision trees, random forests, and artificial neural networks, are utilized in an innovative application to forecast thermal transfer variables. These frameworks possess the ability to scrutinize extensive datasets derived from empirical investigations and simulations to anticipate results with considerable precision, as evidenced by their application in forced convection thermal transfer with nanoparticles, attaining precisions of up to 94% [122,123]. ML models have been used effectively to predict heat exchanger parameters, offering robust performance in estimating thermodynamic properties and flow regimes. These models are built on experimental or numerical data, improving the accuracy of performance predictions and design optimizations [124]. AI/ML models, particularly artificial neural networks (ANNs), have been developed to predict combustion processes in internal combustion engines. These models utilize historical performance data to characterize complex multiphysics flow dynamics, heat transfer, and chemical kinetics, achieving high prediction accuracy [125]. ML techniques are applied to predict thermophysical characteristics and heat-transfer rates in nanofluid systems, aiding in understanding and optimizing these advanced cooling technologies [126]. Deep learning, a subset of ML, is particularly effective in handling complex datasets, and has been applied to predict temperature fields in casting processes with an accuracy of 94.5% [127]. It also facilitates the rapid prediction of thermal displacement in metal additive manufacturing, showing a strong correlation with high-fidelity models [128].

AI methods, including DL, enable real-time simulations and analysis of heat-transfer processes. For example, a deep learning model can calculate temperature variations in less than 0.04 s, providing instant results for online thermal monitoring [129]. Combining different AI techniques, such as machine learning and expert systems, can enhance the flexibility and performance of heat-transfer simulations. These hybrid models are capable of integrating domain expertise with data-driven insights to optimize performance and diagnostics [130]. ML has been integrated into topology optimization for heat conduction problems, significantly reducing computation time by up to 70% while maintaining high-fidelity solutions. This approach is particularly beneficial for cooling designs in electronics, for which rapid simulation results are crucial [131]. AI techniques, such as particle swarm optimization and ANNs, are used to optimize thermal energy storage systems, improving their design and control. These methods improve energy efficiency and reduce environmental impacts by optimizing system performance [132].

AI models significantly reduce computational costs by offering low-fidelity frameworks that can be scaled to high-fidelity models for increased accuracy. This methodology reduces the need for comprehensive numerical simulations, thus saving time and resources [128]. Using machine learning algorithms, scholars can formulate diminished-order models that elucidate intricate thermal transfer phenomena, making them more manageable and computationally efficient [133]. This is particularly beneficial for large-scale thermal systems, for which traditional methods may be computationally prohibitive.

Figure 16 presents a summary of the individual topics discussed in relation to ML and AI.

While machine learning and artificial intelligence provide considerable advantages in thermal transfer simulation, obstacles such as data integrity, assimilation with pre-existing frameworks, and computational expenses persist. Addressing these issues requires optimized computational methodologies and sophisticated sensor technologies [130]. Furthermore, the potential for multiple solutions in inverse problem solving using deep learning poses a challenge that requires further research to refine these models [134]. Despite these challenges, ongoing advances in AI and ML continue to drive innovation in heat-transfer modeling, promising more efficient and sustainable solutions in the future. Furthermore, AI, ML, and DL are very fishable for ROM (see below).

3.3. Reduced-Order Modeling (ROM)

ROM constitutes a sophisticated computational methodology employed to streamline intricate simulations, rendering them viable for immediate applications, especially within the domain of heat transfer. ROMs accomplish this by diminishing the dimensionality of the problem, which leads to a substantial reduction in the computational resources necessary while preserving an acceptable degree of accuracy. This methodology is particularly advantageous in situations where swift simulations are imperative, such as in real-time control systems and digital twins. The following sections offer a comprehensive overview of various ROM methodologies and their implementations in the context of heat transfer. The first ROMs included the use of simple approximation methods, a tactic widely used in industrial applications even in the first half of the twentieth century.

ROM begins by defining the high-fidelity model, typically a detailed simulation using methods like finite element or finite volume, and identifying the governing equations (e.g., Navier–Stokes for fluid flow). Next, data collection involves performing simulations or experiments to generate training data, capturing key states of the system such as velocity, pressure, or displacement fields. The basis for dimensionality reduction is then chosen by using techniques like Proper Orthogonal Decomposition (POD) or Dynamic Mode Decomposition (DMD) to extract dominant features. The governing equations are projected on a reduced basis, resulting in a low-dimensional system of ordinary differential equations (ODEs) or algebraic equations [135]. This forms the reduced-order model, which operates efficiently in a reduced space. The model is validated against the high-fidelity model or experimental data to ensure accuracy, speedup, and stability. Once validated, the ROM is deployed for fast simulations, optimization, control design, or real-time applications [136]. The process is iterative, with updates to the model or reduced basis to improve accuracy or adapt to new scenarios, ensuring that ROM remains effective and reliable [137]. The flowchart of steps in the ROM is presented in Figure 17.

These methods involve modifying the governing equations of the system. A common approach is the Galerkin projection, which projects the full-order model (FOM) onto a reduced subspace. This technique is effective in applications such as the thermal block problem, in which it has been shown to maintain accuracy while reducing computational costs [138,139].

Recent advancements have led to the emergence of non-intrusive ROM techniques, which do not require modifications to the governing equations but instead rely on data-driven methods to construct reduced models. Physics-Informed Neural Networks (PINNs) and Autoencoder-based ROMs are among the most prominent of these techniques. PINNs incorporate physical constraints into neural network architectures, allowing them to model complex heat-transfer problems while preserving physical laws. This approach eliminates the need for explicit equation reformulation and has demonstrated advantages in solving parameterized problems and cases with limited training data. Autoencoder-based ROMs, on the other hand, utilize deep learning frameworks to extract dominant features from high-fidelity simulations, reducing dimensionality while retaining predictive accuracy. These methods are particularly useful for large-scale, nonlinear thermal systems, cases where traditional ROMs may struggle to maintain efficiency and accuracy [140,141].

While non-intrusive ROMs offer significant advantages in scalability and flexibility, they also introduce challenges. Traditional ROMs, such as Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD), are computationally efficient, mathematically well established, and require minimal training data. However, they may struggle with high-dimensional nonlinear systems and often necessitate intrusive modifications to the governing equations. In contrast, non-intrusive ROMs excel in handling complex, data-driven problems without requiring equation reformulation, making them ideal for modern AI-enhanced simulations. However, their main limitations include the need for extensive training datasets and the high computational costs of neural network-based approaches.

ROMs have been effectively utilized to replicate thermally interconnected flow phenomena, including natural convection within a thermally energized cavity. These mathematical constructs are capable of achieving speed improvements exceeding 300 times in comparison to Full Order Models (FOMs), making the former appropriate for applications that require real-time processing [138]. Data-driven ROMs have been formulated for the analysis of convective heat-transfer phenomena within porous media, employing methodologies such as POD and Radial Basis Function (RBF) interpolation techniques. These advanced models have the ability to accommodate parametric uncertainties and have shown substantial improvements in computational efficiency [142]. The integration of machine learning with ROMs, such as in the FastSVD-ML-ROM framework, enhances the ability to handle large-scale nonlinear problems. This approach uses techniques such as singular value decomposition and convolutional autoencoders for dimensionality reduction, allowing real-time simulations in complex systems such as blood flow and fluid dynamics around cylinders [143].

Although ROMs offer significant computational advantages, they also present challenges, particularly in maintaining accuracy under varying conditions. For example, handling temperature-dependent boundary conditions in thermal systems can be complex, requiring advanced reduction strategies such as the Iterative Rational Krylov Algorithm [144]. Additionally, the choice between intrusive and non-intrusive methods depends on the specific application and the availability of system equations [140]. With the increasing role of machine learning in ROM development, hybrid approaches that combine physics-based reduction with data-driven techniques are expected to play a critical role in future heat-transfer simulations. In summary, ROMs establish a comprehensive framework for the simplification of intricate heat-transfer simulations, making them suitable for real-time implementations. Nevertheless, the selection of methodology, be it intrusive or non-intrusive, hinges upon the requirements and limitations of the application in question. As ROMs progressively advance, especially with the incorporation of machine learning techniques, their relevance and effectiveness in real-time contexts are expected to increase significantly. Also, ROM has been applied to an online control system that optimizes the design of the rolling schedule in view of the flatness of the plate. The thermal roll profile during hot rolling was modeled by FEM and then the results were approximated using a time series function or different equations with their numerical solutions [145].

ROM is used to model the dispatch of thermal power in nuclear power plants, allowing efficient heat distributions within industrial processes while reducing carbon emissions. The reduced-order model simplifies complex interactions within the plant, allowing scalability and integration into larger-power models of the system [146]. ROM is applied to the construction of thermal models to optimize control actions during demand–response events. By reducing the order of the model, computational costs are minimized while maintaining model fidelity, which is essential for real-time applications in building energy management [147]. The methodology allows for the automatic identification of key parameters, ensuring that the reduced models accurately capture the thermal dynamics of buildings. In CSP systems, ROM is used to optimize the design and operation of tower receivers and their cavities. The method reduces computational time by orders of magnitude, allowing efficient simulation of heat exchange processes under varying conditions [148]. This efficiency is paramount to enhancing the thermal efficacy of Concentrated Solar Power (CSP) systems, which operate under elevated thermal conditions. ROM is used in district heating networks to efficiently predict evolution of fluid temperature. Reduced models provide accurate simulations of temperature responses, facilitating the design and control of heating systems with significantly reduced computational costs [149]. In battery thermal management systems (BTMS) for electric vehicles, ROM significantly reduces the computational loads of CFD simulations, which are typically resource intensive. Using projection-based methods, ROM can predict transient thermal responses with high accuracy, facilitating rapid design and control optimizations [150].

Figure 18 presents a summary of the individual topics discussed in relation to the ROM.

3.4. High-Performance Computing (HPC)

HPC assumes a crucial position in expansive thermal simulations by offering the requisite computational capabilities to accurately model intricate thermodynamic phenomena with exceptional precision and operational efficiency. Such simulations are imperative in numerous disciplines, including engineering thermodynamics, fluid dynamics, and civil engineering, in which they facilitate the comprehension and forecasting of system behavior under various thermal conditions. HPC allows for the management of extensive datasets and sophisticated models, which are indispensable for achieving precise simulations. The following sections will explore the specific functions and advantages of HPC in large-scale thermal simulations.

HPC allows for the execution of molecular dynamics (MD) and Monte Carlo (MC) simulations, which are essential for investigating thermodynamic problems at the molecular level. These simulations provide insight into phenomena such as droplet coalescence and the development of potential models for thermodynamic properties, such as the Tang–Toennies potential for argon [151].

Advancements in HPC have been significantly influenced by developments in cloud computing and parallel processing, particularly in the realm of thermal management. As computational demands increase, effective thermal management becomes crucial to maintaining system reliability and efficiency. Recent research highlights various strategies and technologies that address these challenges, focusing on liquid cooling, energy-efficient management, and the integration of HPC with cloud computing.

Liquid cooling has emerged as a key technology for managing the thermal output of high-performance systems. For example, the use of liquid cooling in CoWoS (Chip on Wafer on Substrate) systems has demonstrated significant improvements in thermal resistance, with direct liquid cooling achieving junction-to-ambient thermal resistance of 0.055 °C/W at a flow rate of 40 mL/s [152]. A study of two-phase direct-to-chip liquid cooling systems showed that these systems could handle high thermal demands effectively, achieving a Tcase as low as 56.4 °C, which is crucial to maintaining performance in data centers [153].

The Two-Time-Scale Control (TTSC) approach optimizes energy consumption in HPC data centers by dynamically adjusting processor frequencies and task assignments based on thermal conditions. This method has been shown to improve energy efficiency while maintaining thermal constraints [154]. Virtualized HPC cloud infrastructures benefit from autonomic management strategies that use techniques such as voltage scaling and VM migration to manage thermal loads. These methodologies guarantee the optimization of energy utilization while preserving quality of service (QoS) through their ability to adapt to fluctuating thermal environments [155].

The amalgamation of HPC with cloud computing presents scalable and economically efficient solutions for performing intricate simulations and analyses. Cloud platforms provide on-demand resources, mitigating the need for substantial capital expenditures in tangible infrastructure [156]. Advanced VLSI design methodologies, including microfluidic and thermoelectric cooling, are crucial for managing thermal loads in dense circuits. These innovations support the development of AI-specific hardware and other specialized processors, further improving HPC capabilities [157].

Figure 19 presents a summary of the individual topics discussed in relation to HPC.

4. Applications of Numerical Modeling in Thermal Management

This section presents selected applications from among many others and is not a complete list of possible applications.

4.1. Electronics Cooling

Modeling serves an essential function in the governance of thermal dynamics within microelectronics, notably for central processing units (CPUs), graphics processing units (GPUs), and power electronics, by facilitating the design and enhancement of proficient cooling mechanisms. As electronic apparatuses advance in capability and miniaturization, the difficulty of effectively dissipating heat becomes increasingly evident. Modeling offers a systematic approach to forecasting thermal performance, refining cooling methodologies, and ensuring the dependability and efficacy of electronic components. This discourse examines the significance of modeling in thermal management, with an emphasis on CPUs, GPUs, and power electronics.

Thermal models allow for the prediction of temperature distributions and heat flows within electronic devices, which is essential for designing effective cooling solutions. These models help to understand the thermal behavior of complex systems, such as 3D circuits and power electronics [158]. By simulating various cooling scenarios, thermal models enable the optimization of cooling designs, ensuring that devices operate within safe temperature limits. This is especially significant for high-power density utilizations such as Central Processing Units (CPUs) and Graphics Processing Units (GPUs), for which effective thermal dissipation is paramount [159]. For devices experiencing time-varying workloads, such as microprocessors and power electronics, transient thermal models are crucial. These models help to design cooling solutions that can respond quickly to temperature fluctuations, ensuring the reliability and performance of the device [160].

Equation-Based Modeling (EBM) enables the amalgamation of thermal models with conventional 3D chip simulators, permitting an extensive simulation of on-chip thermal occurrences and heat dissipation mechanisms. This methodology has been a transformative factor in addressing the thermal difficulties of nanoscale integrated circuits [161]. Modeling facilitates the advancement and assessment of sophisticated cooling technologies, such as microchannel heat exchangers, phase transition substances, and thermal conduits. These technologies are simulated to evaluate their duration of thermal response and suitability for particular applications [162].

The difficulties associated with the escalation of power densities and the trend toward miniaturization in electronic devices are substantial, as these factors result in heightened thermal burdens that conventional cooling techniques find arduous to control. Sophisticated CFD and ML methodologies are currently being employed to enhance cooling solutions, effectively addressing these issues by improving the efficiency and efficacy of thermal management systems. This analysis investigates the obstacles presented by elevated power densities and miniaturization, as well as the application of advanced CFD and ML methodologies in the optimization of cooling solutions.

As electronic devices become more compact, the power density increases, leading to significant thermal management challenges. For example, high-performance processors can reach power densities of 1–2 kW/cm², creating localized hot spots that degrade performance and reliability [163]. In integrated circuits, rapid temperature swings and sub-millimeter hot spots necessitate sophisticated cooling strategies. Traditional methods often do not address these rapid and localized thermal changes effectively [161]. Miniaturization of components limits the space available for cooling solutions, necessitating innovative designs that can fit within these constraints while still providing effective thermal management [164].

CFD models are crucial to understanding and optimizing the cooling systems of high-power devices. For example, CFD analysis of electric motors has been used to identify and mitigate overheating problems by optimizing airflow and reducing temperature by up to 120 K [165]. The combination of traditional 3D chip simulators with heat dissipation models in EBM languages allows for a comprehensive simulation of on-chip thermal phenomena, facilitating better cooling assessments [161].

Artificial Neural Networks (ANNs) serve as surrogate models to predict thermal resistance and pressure drop, allowing the rapid optimization of cooling systems for high heat-flux applications [166]. DL models, specifically convolutional neural networks (CNNs), are used to forecast cooling effectiveness and improve design variables, considerably reducing computational demands compared to conventional CFD simulations [167]. ML workflows can rapidly explore extensive design spaces, optimizing cooling solutions for power modules and heat sinks. This approach has led to significant improvements in thermal performance and cost efficiency [168].

Although advanced CFD and ML models offer promising solutions for the challenges of increasing power densities and miniaturization, they also introduce new complexities. Integration of these models requires significant computational resources and expertise in both fluid dynamics and machine learning. Additionally, the rapid pace of technological advancement means that cooling solutions must continually evolve to keep up with the new device architectures and power profiles. Despite these challenges, the potential for improved thermal management through these advanced models is substantial, offering pathways to more efficient and reliable electronic devices.

4.2. Energy Systems

Heat-transfer modeling plays a crucial role in optimizing the performance of renewable energy systems such as solar thermal panels, wind turbines, and geothermal setups. By improving the efficiency of heat-transfer processes, these models help maximize energy production, reduce costs, and improve the overall sustainability of renewable energy systems. The integration of advanced materials, innovative design strategies, and sophisticated control mechanisms is one key to achieving these optimizations. Below, the specific contributions of heat-transfer modeling to each type of renewable system are discussed.

The use of Hybrid Nanofluids (HNFs), which combine nanoparticles with conventional heat-transfer fluids, has been shown to enhance the thermal performance of solar thermal systems. These fluids improve heat-transfer rates due to their superior thermal conductivity, as demonstrated in models incorporating copper and alumina oxide nanoparticles suspended in water [169]. Numerical models have been developed to optimize the cooling of Photovoltaic–Thermal (PVT) systems using half-circular tubes and phase change materials (PCMs). These models help determine optimal tube diameters and flow rates, significantly improving both electrical and thermal efficiency [170]. The use of topology optimization models to design fin structures in thermal energy storage units improves heat-transfer rates, thus improving the efficiency of solar thermal systems [171]. Numerical simulations using 3D models have been used to optimize water-cooled photovoltaic–thermal modules. By varying design parameters, such as the velocity of the coolant and the diameter of the tube, significant improvements in the energy and exergy efficiency have been achieved, with reductions in destroyed exergy of up to 81.47% [172].

Dynamic numerical modeling of systems that integrate wind turbines with solar energy and battery storage has been shown to optimize energy generation and reduce CO₂ emissions. These models help determine optimal operational thresholds and land area requirements for wind turbine modules [173]. In systems combining wind and geothermal energy, heat-transfer modeling helps stabilize power output and optimize the use of resources such as hydrogen storage and fuel cells, factors which are crucial to maintaining efficiency under varying conditions [174].

Heat-transfer models for closed-loop geothermal systems focus on optimizing probe geometry and thermal resistance coefficients. These models are validated through in situ experiments, ensuring accurate predictions of fluid output temperature and the thermal capacity of the system [175]. Computational fluid dynamics (CFD) models are used to optimize geometric parameters, such as pipe diameter and fin spacing, to improve the performance of geothermal heat exchangers. These models help achieve significant improvements in thermal performance by optimizing airflow and heat dissipation [176]. Inverse heat conduction models are used to optimize the placement of heat exchangers in geothermal fields, minimizing adverse effects and enhancing long-term sustainability. This approach ensures efficient heat transfer within and outside exchangers [177].

Combining solar and geothermal energy sources in cogeneration systems involves selecting the optimal heat-transfer fluids and design parameters to maximize energy efficiency. For example, using Therminol 59 as a heat-transfer fluid in parabolic trough solar collectors has been shown to achieve high energy efficiency and net power output [178]. In amalgamated systems that integrate photovoltaic cells and parabolic trough collectors, thermal transfer modeling is employed to enhance parameters such as fluid ingress temperature and solar irradiance intensity. This culminates in increased thermal and electrical exergies, decreased CO₂ emissions, and maximized cost efficiency [179].

Although thermal transfer modeling substantially increases the efficacy of renewable systems, it is imperative to consider the economic and ecological repercussions of these enhancements. For example, the incorporation of sophisticated materials such as nanoparticles and phase change materials (PCMs) can escalate expenses and complexity. Furthermore, the ecological advantages of reduced CO₂ emissions and enhanced energy efficiency must be evaluated against the possible environmental ramifications of the production and disposal of these materials. Consequently, a judicious approach that recognizes both technological innovations and sustainability is crucial for the successful execution of thermal transfer models in renewable energy systems.

The thermal management and temperature control of fuel cells are critical to ensuring efficiency and safety in energy storage systems. Effective thermal management systems (TMS) are essential in order to maintain optimal operating temperatures, an ability which directly impacts the performance, lifetime, and safety of batteries and fuel cells. This answer explores the various strategies and technologies used in battery thermal management and their applications in fuel cells, highlighting the importance of temperature control.

Battery Thermal Management Systems (BTMS) can be categorized into active and passive systems. Active systems require external energy to operate, such as air or liquid cooling, whereas passive systems rely on natural heat dissipation methods such as phase change materials (PCMs) [180,181]. Advanced BTMS designs focus on controlling the temperature at the cell level rather than the pack level. This approach allows for more precise temperature management, reducing temperature gradients and extending battery life [182]. Combining active and passive cooling methods, such as by using PCMs with air cooling, can improve thermal management efficiency. This hybrid approach is particularly effective in managing heat generated during charge and discharge cycles in high-energy-density batteries [180].

Effective thermal management in Proton Exchange Membrane Fuel Cells (PEMFCs) involves managing heat dissipation and temperature distribution to ensure uniform temperature across the cell stack. This is crucial in order to maintain performance and prevent degradation [183]. Integrating thermal energy storage (TES) units with radiators in fuel cell vehicles can reduce the size and weight of the thermal management system, improving vehicle efficiency and range [184]. New concepts, such as the use of metal hydrides for hydrogen storage, offer innovative solutions for thermal management in fuel–cell hybrid electric vehicles. These systems take advantage of the exothermic and endothermic properties of hydrogen absorption and desorption to effectively regulate battery temperatures [185].

Machine learning-based models are being developed to predict and manage battery temperatures more accurately, improving the efficiency of BTMS by optimizing cooling strategies [186]. One novel approach involves the use of electrolyte convection within lithium-ion batteries to improve heat removal and reduce internal temperature gradients, offering potential advantages in rapid temperature regulation and cost reduction [187]. The use of additive manufacturing in the development of BTMS allows greater design flexibility and the integration of latent heat storage materials, which can improve thermal management without significantly increasing the weight or volume of the system [188].

Although the emphasis on battery thermal regulation and fuel cell applications underscores the importance of temperature management for efficiency and security, it is also imperative to consider the obstacles and constraints of these systems. For example, the incorporation of sophisticated cooling technologies can improve the complexity and expense of energy storage systems. Furthermore, the advancement of novel materials and technologies must equilibrate performance enhancements with ecological and economic considerations. As research advances, addressing these obstacles will be crucial to the widespread acceptance of efficient and secure energy storage solutions.

4.3. Automotive and Aerospace

Thermal management modeling needs in automotive and aerospace applications are associated with crucial efforts to optimize engine cooling, braking systems, and climate control. These systems require sophisticated models to ensure efficiency, safety, and performance. In automotive applications, the focus is on engine cooling systems, battery thermal management, drive motor cooling, energy-saving strategies, and integrated thermal management systems. In aerospace, similar principles apply, but with additional considerations for the unique environmental conditions and operational demands. The following sections detail the specific modeling needs for each area.

Models for engine cooling systems, such as those that use silicone oil fan clutches, are essential for the analysis of performance under various control strategies. These models help researchers to understand the impacts of different fan control strategies on coolant temperature and power consumption, leading to optimized control strategies, such as adaptive fuzzy PID controllers, that significantly reduce power consumption [189]. Simulation tools such as KULI are used to model the entire engine thermal management system, including components such as water tanks and fans. These models guide the selection and matching of components to ensure optimal performance in different driving cycles, such as the NEDC [190]. Advanced systems incorporate multiple components, such as high- and low-temperature radiators, EGR coolers, and integrated thermal management valves. These models prevent problems like turbo-boiling and optimize warm-up times for engines and transmission fluids [191]. Although the contexts provided do not directly address braking systems, thermal management in braking is critical, especially in high-performance and heavy-duty vehicles. Models typically focus on heat dissipation and the impact of braking on overall vehicle thermal management.

In electric vehicles (EVs), battery thermal management is crucial for ensuring safety, efficiency, and longevity. Lithium-ion batteries generate significant heat during charging and discharging, necessitating advanced cooling and heating systems to prevent thermal runaway and maintain optimal performance. Common thermal management strategies include air cooling, liquid cooling, phase change materials (PCMs), and immersion cooling, each offering distinct advantages depending on vehicle design constraints. Liquid cooling, in particular, is widely adopted because of its high heat dissipation efficiency, while immersion cooling is gaining interest because of its ability to enhance temperature uniformity and prevent hotspots. Additionally, the integration of heat pumps into EV battery systems allows for improved thermal regulation in extreme environmental conditions.

Thermal management of drive motors is another critical aspect of EV and electric aircraft (E-Aircraft) design. Electric motors generate substantial heat as a result of resistive losses in the windings and iron losses in the stator and rotor cores. Efficient cooling strategies, such as direct liquid cooling, oil-spray cooling, and heat pipe-based thermal management, are employed to maintain motor efficiency and prevent overheating. Furthermore, waste heat recovery systems can be integrated into the thermal management framework to enhance overall energy efficiency. In electric aircrafts, where weight constraints are paramount, novel lightweight cooling materials and advanced thermal interface materials (TIMs) are being explored to improve heat dissipation without adding excessive mass.

In electric vehicles, models for HVAC systems and heat pumps are crucial for maintaining cabin comfort and battery temperature. These models are validated against high-fidelity simulations to ensure accuracy and are designed to be sufficiently computationally efficient for real-time applications [192].

Advanced computational techniques, including CFD and FEM, are pivotal for addressing extreme temperature ranges, lightweight construction, and fuel efficiency. Widely applied in engineering disciplines, these methods enhance performance and design optimization. CFD excels in analyzing fluid flow and heat transfer, while FEM specializes in structural analysis and thermal management. Together, they offer robust solutions to intricate engineering problems.

CFD is used to simulate and optimize vehicle thermal management systems, ensuring effective performance even under extreme temperature conditions. For example, CFD simulations aid in the development of cooling systems that regulate the thermal loads of engines and components, improve fuel efficiency, and reduce emissions [193]. FEM is applied in the process of forming magnesium alloy sheets at elevated temperatures, an application which is crucial to improving the formability of lightweight materials. This method allows accurate predictions of the effects of temperature on material behavior, enhancing the manufacturing process of components that must withstand high temperatures [194].

FEM, combined with topology optimization, is applied to decrease the weight of components while preserving structural integrity. This method is especially advantageous in the automotive and aerospace sectors, in which weight reduction significantly improves fuel efficiency and performance [195]. FEM is used to optimize the designs of lightweight structures, such as vehicle frames and engine components, by simulating static and dynamic performance. This approach ensures that structures meet performance requirements while reducing weight, which is essential in order to improve fuel efficiency [196]. The use of lightweight materials such as magnesium alloys and titanium aluminides is facilitated by FEM simulations, which help to understand the temperature gradients and mechanical properties of these materials under operating conditions [197].

In aerospace, CFD-based shape optimization of heat exchangers is employed to minimize weight and drag, thereby improving fuel efficiency. Optimized designs can achieve significant mass savings, especially under high heat loads, an achievement which is critical for future hydrogen-powered aircraft [198].

Although CFD and FEM present considerable benefits in the regulation of extreme thermal conditions, light configuration, and fuel efficiency, it is crucial to consider the constraints and difficulties associated with these methodologies. For example, the intricacy and computational expense of generating intricate three-dimensional models can be considerable, and the precision of simulations is heavily dependent on the caliber of the input data and the premises established during modeling. Furthermore, amalgamation of these methodologies with other sophisticated design techniques, such as metaheuristic algorithms and optimization strategies, can further enhance their efficacy in addressing complex engineering challenges.

4.4. Industrial Processes

Numerical models play a central role in augmenting the efficacy of heat exchangers, chemical reactors, and other industrial operations by optimizing thermal performance. These models permit intricate simulations and forecasts of thermal behavior, allowing engineers to devise systems that maximize heat-transfer efficiency while minimizing energy consumption and operational expenditures. By incorporating sophisticated computational methodologies such as CFD, ANN, and PINN, these models yield insights into intricate thermal processes and facilitate real-time optimization. Furthermore, green computing strategies are increasingly being integrated into industrial thermal management, focusing on achieving reductions in energy use and carbon emissions while maintaining computational efficiency. The following sections investigate how numerical models contribute to the enhancement of thermal performance in industrial applications.

Numerical models using the Volume of Fluid (VOF) method simulate multiphase flow in Pulsating Heat Pipes (PHPs), revealing phase separation and convective flow patterns that enhance heat-transfer efficiency. ANN models further optimize these systems by predicting volume fractions and wall temperatures with high accuracy, significantly reducing computational time [199]. Numerical simulations of heat exchangers using nanofluids demonstrate improved thermal efficiency. For example, Cu-water nanofluids improve heat transfer by 19.33%, and inner tube rotation increases efficiency by 41.2%. However, these improvements come with an increase in pressure drop and pumping power requirements, which require a balance between performance and operational costs [200].

PINNs are employed to model and optimize the temperature trajectory in tubular reactors, maximizing reactor yields. These models provide a closed analytical form of solutions, offering computational advantages over traditional methods such as the FDM [201]. Additionally, ROM techniques have been introduced as an effective means of reducing computational demand while maintaining high accuracy in industrial simulations. ROM-based approaches in industrial process modeling significantly reduce energy consumption by enabling real-time optimization and reducing the dependency on high-performance computing (HPC) resources, aligning with green computing principles.

ANNs and GAs are used for thermal analysis and multi-objective optimization of heat exchangers, improving efficiency and production quality. These models address complex nonlinear processes, facilitating quick error resolution and process automation [202]. Physics-Informed Deep Learning accelerates the design and optimization of heat exchangers by integrating space decomposition, transfer learning, and advanced linear algebra techniques. It enables the discovery of optimal geometric designs and operating conditions under uncertainty [203].

In addition to the factor of improved computational efficiency, numerical simulations play a key role in energy-saving strategies for industrial thermal management. Waste heat recovery systems, such as regenerative heat exchangers and organic Rankine cycles (ORCs), have been successfully modeled to enhance energy reuse in high-temperature industrial processes like metal smelting and chemical refining. These methods help industries reduce their overall energy consumption and greenhouse gas emissions.

Although numerical models significantly improve the efficiency of heat exchangers and reactors, they also present challenges, such as computational limitations and the need for precise modeling. For example, CFD models in nuclear reactors require careful consideration of boundary conditions and turbulence modeling to ensure safety and reliability [204]. Additionally, the integration of chemical reactions and heat exchange in compact devices, such as micro-reactors, poses unique design constraints and opportunities for innovation. These challenges highlight the importance of continuous advances in numerical modeling techniques aiming to address the evolving needs of industrial processes.

Numerical modeling plays a crucial role in thermal management in various manufacturing processes, particularly thermal processing, additive manufacturing, and other temperature-sensitive applications. These models help to optimize thermal conditions, improve product quality, and enhance process efficiency. The integration of numerical simulations in these fields allows precise control over temperature distributions, which is essential for maintaining the integrity and performance of manufactured components. The applications of numerical modeling in these areas are discussed in detail below.

Numerical modeling in Powder Bed Fusion (PBF) processes is used to simulate thermal management strategies, such as adaptive cooling times, to control the thermal history of metal parts. This approach helps reduce maximum temperatures and manufacturing time, thus improving the mechanical properties and microstructure of parts [205]. Hybrid thermal models using physics-informed neural networks have been developed to predict temperature distributions in additive manufacturing. These models combine experimental data with physical laws to improve prediction accuracy and identify unknown parameters, facilitating real-time process control and iterative design [206]. Numerical models are used to study temperature distributions during layer-by-layer deposition in laser powder-based additive manufacturing. These models help to understand thermal gradients, which are crucial for reducing residual stresses and improving geometric stability [207]. The holistic model, the main algorithm and the simulation results of the PBF processes (including selective laser melting and sintering) associated with one example are presented in a series of publications [208,209,210,211,212]. They present multi-material, multitrack, and multilayer simulations of metallic and non-metallic materials.

Numerical modeling is crucial for improving the efficiency and precision of Chemical Vapor Deposition (CVD). Widely used in industries such as those associated with semiconductors and thin-film coatings, CVD relies on precise thermal management to ensure uniform, high-quality deposition. Modeling thermal and fluid dynamics enables better process control and optimization [213].

Numerical simulations are used to design thermal management systems using PCMs for applications such as battery thermal management. These models help optimize the thermal conductivity and melting points of PCMs to maintain safe temperature ranges during operation [214]. Additive manufacturing techniques are used to create novel thermal management systems for electric machine windings. Numerical models evaluate the effectiveness of PCM in reducing temperature increases, thereby improving the performance and useful life of electric machines [215].

To further improve industrial energy efficiency, new technologies such as solar-assisted thermal systems and hybrid energy storage solutions are being modeled to integrate renewable energy into thermal processes. These approaches help industries transition to carbon neutrality by minimizing dependence on fossil fuel-based thermal management systems. Furthermore, smart energy management systems that use AI-driven predictive modeling are being incorporated to dynamically optimize industrial cooling and heating operations based on real-time energy demand and external environmental conditions.

Numerical models simulate the thermal behavior of lithium-ion battery packs, using composite phase change materials. These models help extend the operating temperature range and improve insulation effects, factors which are critical to battery safety and performance [214]. The additive manufacturing of metal components is being investigated for heat-transfer applications. Numerical modeling is used to simulate fluid flow within channels, highlighting the effects of surface roughness on performance and underscoring the importance of model calibration [216].

Although quantitative simulation presents considerable benefits in thermal regulation, it is crucial to consider the constraints and difficulties associated with these methodologies. For example, the accuracy of numerical simulations can be affected by factors such as surface roughness in additive manufacturing, a consideration which requires model calibration [216]. Additionally, the computational cost and complexity of purely physics-based models can be prohibitive, requiring the development of hybrid models that integrate data-driven approaches [206]. The adoption of green computing strategies, including cloud-based simulation frameworks powered by renewable energy, is increasingly being explored to minimize the environmental impact of large-scale numerical modeling efforts. These methods not only reduce carbon emissions but also improve the accessibility of computational resources used for sustainable industrial development. These considerations highlight the need for ongoing research and innovation to further enhance the capabilities and applications of numerical modeling in thermal management.

4.5. Healthcare and Biomedical Applications

Numerical modeling plays a crucial role in the advancement of thermal management applications in various niche areas, including medical device cooling, hyperthermia treatment modeling, and thermal imaging in diagnostic tools. These applications use numerical simulations to optimize thermal performance, improve device reliability, and improve patient outcomes. The integration of advanced computational techniques, such as machine learning and artificial intelligence, further enhances the precision and efficiency of these models. Below, the applications of numerical modeling in these niche areas are explored in detail.

The prediction of the thermophysical properties of nanofluids, essential for effective cooling in medical devices, is heavily based on advanced numerical modeling. To improve the accuracy and reliability of these predictions, sophisticated machine learning techniques like Artificial Neural Networks (ANN) and Support Vector Regression (SVR) have been integrated into modeling frameworks. These machine learning algorithms analyze extensive datasets, allowing for a deeper and more precise understanding of the behavior and properties of nanofluids under varying conditions. When these predictions are made, the integration of machine learning ensures that the cooling systems in medical devices are optimized to achieve superior efficiency and consistent performance. This advancement not only improves the operational reliability of medical devices, but also supports critical healthcare needs by enabling more effective thermal management solutions [217]. In medical devices powered by batteries, numerical modeling helps design effective Battery Thermal Management Systems (BTMS). For example, thermoelectric coolers (TECs) have been modeled to improve cooling efficiency, ensure safe operating temperatures, and reduce energy consumption [218].

Numerical simulations are crucial in hyperthermia treatment, in which precise control of temperature is necessary to target cancer cells without damaging surrounding healthy tissue. Advanced modeling techniques help to understand the dynamics of heat transfer and optimize treatment protocols [219].

Numerical modeling aids in analyzing the thermal characteristics of diagnostic tools, such as thermal imaging cameras. By simulating heat-transfer processes, these models help improve the accuracy and reliability of thermal imaging, which is critical for diagnostic purposes [220]. The integration of numerical simulations with artificial intelligence can optimize cooling systems in diagnostic tools, ensuring stable operation and preventing overheating, which could otherwise lead to inaccurate readings [3].

Although numerical modeling offers significant advances in thermal management for medical applications, it is important to consider the limitations and challenges associated with these techniques. For example, the complexity of biological systems and the variability in patient-specific conditions can pose challenges in accurately modeling hyperthermia treatments. Furthermore, the integration of AI and machine learning in numerical modeling requires substantial computational resources and expertise, which may not be readily available in all settings. Despite these challenges, the continued development and refinement of numerical modeling techniques holds promise for further enhancing thermal management in medical applications.

5. Environmental Impacts of Computational Modeling for Heat Transfer

5.1. Energy and Resource Demands

The environmental impacts of computational modeling for heat transfer, particularly through HPC, are significant because of the energy consumption and carbon emissions associated with these processes. As computational simulations become more integral to scientific and engineering advancements, understanding and mitigating the environmental footprint of these simulations becomes crucial. This involves looking at both operational energy consumption and carbon emissions from hardware manufacturing and infrastructure. Recent studies estimate that HPC systems used for numerical simulations can consume between 10 and 50 MW of power, depending on the scale of the computing infrastructure, with CO₂ emissions ranging from 1.5 to 3.0 kg per kWh in regions powered by fossil-fuels [221,222]. For example, a single large-scale CFD simulation can generate between 200 and 500 kg of CO₂ emissions, depending on complexity and duration [222].

HPC systems are characterized by their substantial energy demands, which require considerable power for computational processes and thermal management. Although their energy consumption may not be directly comparable to those of energy-intensive industrial processes such as fuel combustion for heating, HPC systems still contribute significantly to carbon emissions, particularly when powered by fossil fuel-based energy sources. For example, the Fugaku supercomputer, ranked No. 1 in the TOP500 list at the International Supercomputing Conference (ISC) High Performance 2020 (TOP500, 2020), had a power consumption of 28.3 MW at that time, which was equivalent to the annual power consumption of 70,000 households, assuming an average household power consumption of 400 W [223]. This impact is increasingly relevant as computational workloads grow, emphasizing the need for energy-efficient solutions in high-performance computing [221,222]. The operational energy use associated with HPC systems is experiencing a decline attributable to significant progress in the algorithms, software, and hardware that optimize performance and energy efficiency. However, aggregate carbon emissions continue to increase as a result of the expanding scale and increasing demand associated with HPC resources [224].

As highlighted in the introduction, the growing energy demands of AI-driven computing and data centers are exacerbating these concerns. The IEA’s “Energia 2024” report warns that electricity consumption for AI and cryptocurrency could double by 2026, reaching levels equivalent to Japan’s total energy use. Projections indicate that U.S. data centers may consume 8% of the nation’s electricity by 2030 (up from 3% in 2022), straining power grids. This has driven major tech companies, including Microsoft and Amazon, to secure nuclear energy sources, though regulatory concerns—such as FERC’s recent rejection of Amazon’s nuclear power deal—underscore the challenges of balancing computational growth with grid stability.

The energy consumption of machine learning (ML) models varies significantly according to architecture and infrastructure choices. Large but sparsely activated deep neural networks (DNNs) can use less than one-tenth of the energy of dense models, while maintaining accuracy. Additionally, cloud data centers are 1.4–2× more energy-efficient than standard data centers, and ML accelerators (e.g., TPUs, specialized GPUs) are 2–5× more efficient than general-purpose hardware. These optimizations can reduce the carbon footprint of ML workloads by up to 100–1000×, highlighting the importance of energy-efficient architectures, low-carbon computation scheduling, and high-performance AI hardware [225].

Carbon emissions from HPC systems are not only due to operational energy consumption but also significantly result from hardware manufacturing and infrastructure. The production of HPC components and the construction of data centers contribute substantially to their carbon footprints [224]. The carbon embodied in a large-scale HPC system, including semiconductor fabrication and facility infrastructure, can contribute up to 20% of its total lifecycle emissions, underscoring the importance of sustainable hardware design [226]. The regional intensity of carbon plays a role in the carbon footprint of HPC systems, as the components of the energy source mix (renewable vs. non-renewable) affect the emissions associated with power generation [222].

The implementation of energy-efficient algorithms, along with the incorporation of renewable energy sources into high-performance computing operations, represents a progressive approach to mitigating the ecological footprint associated with computational modeling. Innovative cooling technologies and material recycling can also contribute to the development of computing practices which are more sustainable [227]. Recent advances in liquid immersion cooling have demonstrated significant energy savings in high-performance computing (HPC) data centers. Studies indicate that immersion cooling can reduce energy consumption by approximately 50%, compared to traditional air cooling methods [228]. Additionally, the implementation of dynamic workload scheduling based on carbon intensity forecasts has been shown to effectively reduce emissions. For example, the CASPER system, a carbon-aware scheduling and provisioning framework, has achieved substantial reductions in carbon emissions for distributed web services by optimizing workload distribution in response to real-time carbon intensity data [229]. The principles of GREENER provide a framework for environmentally sustainable computational science, highlighting the need for greater awareness and transparency, and improved estimation of environmental impacts [230]. Tools such as Green Algorithms provide a framework for estimating and reporting the carbon footprint of computational tasks, helping users minimize unnecessary CO₂ emissions and promote greener computation [231].

In CFD, the carbon footprint of simulations can be substantial, especially for “hero” calculations that require significant computing resources. However, efforts to avoid redundant calculations relative to turbulence databases have shown the potential to reduce CO₂ emissions by significant margins [221]. One such initiative is the Johns Hopkins Turbulence Database (JHTDB), which provides centralized access to precomputed turbulence data, eliminating the need for repeated, high-energy-consuming simulations. Using shared turbulence datasets instead of performing direct numerical simulations (DNS) for each new study, researchers can significantly reduce computational requirements and energy consumption.

Estimates suggest that using turbulence databases such as JHTDB could potentially reduce CO₂ emissions by approximately one million metric tons [221].

In astrophysics, the environmental impact of computing exceeds that of telescope operations, highlighting the need for optimized code and efficient hardware usage to mitigate greenhouse gas emissions [232].

Although the ecological implications of computational modeling for thermal transfer are an increasingly pressing issue, there exist avenues for enhancement. Advancement of sustainable computing methodologies, including the implementation of energy-efficient algorithms and the use of renewable energy sources, can significantly mitigate carbon emissions related to HPC systems. Integrating AI-driven surrogate modeling techniques, such as physics-informed neural networks (PINNs), can significantly reduce computational demands, while maintaining accuracy. For instance, the Separable PINN (SPINN) approach has demonstrated a 62-fold decrease in wall-clock time and a 1,394-fold reduction in floating-point operations (FLOPs) when solving multi-dimensional partial differential equations (PDEs), compared to traditional methods. In one notable example, SPINN solved a chaotic (2 + 1)-dimensional Navier–Stokes equation in 9 min, whereas previous methods required 10 h on a single GPU, achieving comparable accuracy. These advances highlight the potential of PINNs relative to improved computational efficiency in complex simulations [233]. Furthermore, tools such as Green Algorithms offer critical insights into the carbon footprint of computational activities, thereby empowering researchers and practitioners to make judicious choices regarding their computing methodologies. As the requirement for HPC escalates, collaborative initiatives among academic institutions, industry stakeholders, and policymakers will be paramount in nurturing an environmentally sustainable digital future.

5.2. Environmental Benefits Through Optimization

Computational modeling for heat transfer plays a crucial role in the improvement of energy efficiency by optimizing thermal systems in various sectors. These models allow precise simulations and predictions, enabling the design and operation of systems that minimize energy consumption and waste heat. A comparative lifecycle assessment of different modeling techniques shows that AI-assisted optimization of heat exchangers, particularly through maldistribution minimization techniques, can improve energy efficiency by 20–40% compared to traditional CFD-based approaches, as demonstrated in compact absorption cooling applications [234]. By integrating AI-driven methodologies, such as genetic algorithm (GA)-based optimization, flow maldistribution can be significantly reduced, improving heat exchanger performance while lowering overall energy consumption. By leveraging advanced numerical techniques, computational models significantly contribute to reducing the environmental impact of thermal systems. The following sections explore how these models optimize thermal systems, leading to improved energy efficiency and reduced waste heat.

Computational models are essential for the optimization of thermal systems in manufacturing, electronics, and heat rejection applications. These systems are complex due to their variable material properties and multiscale phenomena, which require detailed modeling and simulation to achieve accurate results. When optimizing these systems, energy and material consumption can be reduced, enhancing productivity and product quality while minimizing environmental impact [235]. The integration of CFD with ROMs allows rapid and accurate analysis of complex systems. For example, a hybrid CFD and ROM approach has been developed for vehicle thermal analysis, achieving a speedup ratio of up to 27 times without compromising accuracy, with relative errors below 0.35% and absolute errors less than 4 K [236]. In the domain of cold storage, the application of Computational Fluid Dynamics (CFD) in conjunction with various modeling methodologies serves to refine temperature and velocity distributions, thus mitigating thermal burdens and improving energy efficiency. This methodology not only supports food safety but also culminates in considerable energy conservation [237].

Numerical modeling is used to optimize energy consumption in buildings by reducing heat losses and improving thermal comfort. Techniques such as the assessment of the life cycle and the integration of renewable energy sources, such as solar panels, further improve energy efficiency and reduce CO₂ emissions [238]. In industrial systems, computational models help design heat exchanger networks that minimize environmental impact. When optimizing the temperature difference (ΔT min), waste release is significantly reduced, demonstrating the dual benefits of economic and environmental optimization [239].

In wastewater treatment, heat exchangers with heat pipes are modeled to efficiently recover thermal energy. This method not only saves fossil fuels and reduces CO₂ emissions but also exploits a sustainable energy source, showcasing the environmental benefits of optimized heat-transfer processes [240]. Latent Heat Thermal Energy Storage (LHTES) systems utilize phase change materials to store and release thermal energy efficiently. Advanced modeling approaches help to understand phase transitions and optimize these systems for better energy management, thus reducing ecological impacts [241].

Although computational modeling significantly improves energy efficiency, it also presents challenges, such as the need for high computational power and expertise in scientific domains. The complexity of the models and the potential for errors in mesh generation and simplification must be carefully managed to ensure accurate results [242]. Additionally, the development of open-source model structures for energy systems with sector coupling is crucial to achieving decarbonization targets and promoting transparency and reproducibility in energy modeling [243].

In conclusion, the use of computational modeling for heat transfer constitutes a formidable instrument for the optimization of thermal systems, resulting in enhanced energy efficiency and the reduction of waste heat across diverse sectors. However, the complexity and computational requirements inherent in these models require meticulous oversight and specialized knowledge to fully realize the prospective advantages of these models.

5.3. Sustainability and Lifecycle Assessment

The environmental impacts of computational modeling for heat transfer, particularly in the context of lifecycle assessment (LCA) of thermal management solutions, are multifaceted. Numerical modeling plays a crucial role in optimizing thermal management systems, leading to reduced material and energy costs. One key approach to reducing the carbon footprint of computational modeling is the integration of carbon-aware scheduling frameworks, which dynamically adjust workloads based on grid energy availability. Studies have shown that the incorporation of energy storage solutions, such as batteries, into these frameworks can lead to emission reductions ranging from 15% to 65%, depending on regional energy sources [244]. This optimization is achieved through the development of advanced materials and systems that improve energy efficiency and minimize environmental impacts. The following sections explore the lifecycle assessment of various thermal management solutions designed through numerical modeling, focusing on their environmental benefits and challenges.

The lifecycle assessment of solar-thermal systems reveals that careful selection of components can significantly improve environmental performance. For example, using lithium-ion batteries and copper–indium–selenium (CIS) solar collectors can reduce environmental impacts in several categories. Computational modeling help optimize these systems by simulating different configurations and materials, leading to reductions in emissions and waste production [245]. The development of thermal insulation blocks from secondary raw materials, such as glass and plastic, demonstrates the potential to reduce environmental impacts through optimized production processes. Numerical modeling helps design production lines that minimize energy consumption and material waste, achieving up to 30% reduction in environmental impacts [246]. The lifecycle assessment of polystyrene thermal insulation materials highlights the importance of evaluating environmental impacts throughout the lifecycle of the product. This includes reducing heat losses, sourcing raw material, and considering the processing and disposal methods. Expanded polystyrene boards, particularly those containing graphite, have shown lower environmental impacts due to reduced polymer content and enhanced structural integrity [247]. The use of phase change materials, such as paraffin wax, in thermal energy storage systems benefits from computational modeling to optimize heat-transfer processes. This approach reduces energy and exergy losses, thus improving system efficiency and sustainability indices [248].

The computational simulation of an industrial ceramic furnace has shown that optimization of its thermal dynamics can result in an 83% reduction in energy utilization and an 87.36% reduction in carbon dioxide emissions. This underscores the ability of numerical modeling to improve energy efficiency and mitigate environmental repercussions in industrial contexts [249].

The use of PCMs for latent heat storage in buildings is a promising solution for reducing energy consumption. However, the environmental impacts of PCMs are not always favorable. Although they offer a high density of energy storage, their application in building structures can lead to increased embodied environmental impacts, particularly when using organic PCMs based on fossil fuels [250]. A comprehensive LCA of a solar–LHTES–PCM system revealed that while the system has high environmental impacts in several categories, extending its lifetime to 40 years can significantly reduce these impacts, making it a more sustainable option compared to traditional heating methods [251].

Although numerical modeling and the lifecycle assessment of thermal management solutions offer significant potential for the reduction of environmental impacts, challenges remain. The accuracy of LCA studies is often limited by the availability of precise data, particularly for emerging technologies such as PCMs. In addition, the environmental benefits of these solutions must be weighed against their economic feasibility and the potential for unintended consequences, such as increased embodied impacts. Future research should focus on improving the accuracy of data and exploring innovative materials and systems that balance environmental and economic considerations.

5.4. Challenges and Future Needs in Green Computing

The environmental impacts of computational modeling for heat transfer are significant, primarily due to the energy-intensive nature of large-scale simulations and the operation of data centers. These activities contribute to high carbon emissions and electronic waste, which pose challenges to sustainability. However, there are ongoing efforts to mitigate these impacts through green computing initiatives and sustainable algorithm designs. For example, the Massachusetts Green High Performance Computing Center (MGHPCC) in Holyoke, Massachusetts, sources approximately 94% of its energy from carbon-neutral sources, with hydroelectric power accounting for about 66.7% of its annual consumption [252]. This response explores the current challenges and future directions in this domain.

Data centers, which facilitate computational modeling, represent substantial consumers of energy. They contribute to a considerable fraction of worldwide electricity consumption, with forecasts suggesting that Information and Communication Technology (ICT) may use up to 21% of global energy by the year 2030. The rapid growth of computing technologies leads to increased electronic waste, which exacerbates environmental concerns. This waste includes obsolete hardware and components, which are often not recycled efficiently [253]. The operational energy consumption of these centers contributes to their carbon footprint, despite their improvements in power efficiency. The carbon emissions from data centers are primarily due to the reliance of these centers on non-renewable energy sources. The manufacturing and infrastructure associated with computing hardware also contribute significantly to emissions [224]. The data centers which support these simulations consume vast amounts of energy and water and generate electronic waste. Their rapid growth exacerbates environmental concerns [254].

Initiatives like SustainDC use machine learning to optimize data center operations, including workload scheduling and cooling optimization. These approaches aim to reduce energy consumption and improve sustainability by using advanced algorithms [255]. A comprehensive approach to green computing involves designing and manufacturing computing devices with a reduced environmental impact. This includes using materials which are less hazardous and maximizing the efficiency of the product lifecycle [256,257].

Developing algorithms that require less computational power can reduce energy consumption. These algorithms are designed to optimize performance while minimizing resource use. Advanced cooling systems are being implemented to enhance energy efficiency in data centers. These systems reduce the need for traditional air conditioning, which is energy-intensive [227]. The environmentally sustainable computing (ESC) framework offers a holistic methodology for embedding environmentally conscious principles throughout the lifecycle of computing systems, covering all phases, from design to operational execution [258].

6. Challenges in Numerical Modeling for Heat Transfer

Numerical modeling related to thermal transfer constitutes a substantial challenge to the effort to achieve a harmonious balance between elevated precision and manageable computational resources. This trade-off is of paramount importance as it influences the practicality and effectiveness of simulations across a spectrum of applications, encompassing industrial processes and environmental research. The intricate nature of thermal transfer mechanisms, including conduction, convection, and radiation, frequently necessitates the use of advanced models that may be computationally demanding. However, progress in computational methodologies and technological innovations presents promising avenues for addressing these challenges.

Heat-transfer processes often involve multiple mechanisms, including conduction, convection, and radiation, which can interact in complex ways. For example, radiative transfer in absorbing and scattering media, such as gases with suspended particles, requires detailed calculations across a wide spectral range, significantly increasing computational demands [259]. Dynamic models, including those used in heat exchange systems, can be refined to ensure adequate precision while concurrently reducing the demand for computational resources. For example, parameterizable models derived from analytical solutions can attain elevated computational speeds while maintaining minimal error rates [260]. The introduction of dimensionless metrics for the purpose of quantifying the concordance between theoretical models and empirical data serves to enhance the evaluation of simulation accuracy. This methodology facilitates a more nuanced understanding of the compromises inherent in model selection [261].

Numerical modeling in the domain of heat transfer constitutes a multifaceted discipline that encounters numerous challenges, especially in the endeavor to accurately replicate real-world phenomena such as transient behaviors and fluctuating boundary conditions. These challenges emerge from the intrinsic complexity associated with heat-transfer mechanisms, which frequently involve various transport processes and necessitate precise modeling of dynamic boundary conditions. The constraints observed in contemporary models stem primarily from the obstacles associated with the accurate simulation of these transient and variable conditions, which is imperative to produce realistic and reliable forecasts.

Fluid–Structure Interaction (FSI) simulations necessitate precise coupling between Computational Fluid Dynamics (CFD) and Finite Element Method (FEM) models to accurately capture the interplay between fluid flows and structural deformations. A significant challenge in FSI is ensuring stable and efficient data transfer between CFD and FEM solvers, especially during large deformations or transient events. Two primary coupling strategies are employed:

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Partitioned (or loosely coupled) approaches: These solve the fluid and structural equations separately, iteratively exchanging boundary conditions. Although offering flexibility, they require robust interpolation schemes to transfer data such as pressure, displacement, and thermal loads between solvers [262].

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Monolithic (or strongly coupled) approaches: These solve fluid and structural equations simultaneously within a unified framework, enhancing stability but increasing computational complexity [263].

In partitioned methods, implicit coupling with sub-iterations can improve stability, but at a higher computational cost, whereas explicit coupling reduces computational load but demands careful time-stepping to prevent numerical instabilities. Techniques such as the Arbitrary Lagrangian–Eulerian (ALE) formulation and Immersed Boundary Methods (IBM) are utilized to manage large deformations without frequent remeshing [264,265].

The simulation of phase change processes such as solidification, evaporation, and boiling presents substantial computational challenges due to rapid property variations and moving phase boundaries. Accurate modeling of these phenomena is crucial for applications ranging from materials processing to energy systems. Several numerical methods have been developed to address these challenges:

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Sharp-interface methods: Techniques such as Front Tracking and Level-Set Methods explicitly track phase boundaries, providing precise interface representation but often requiring fine spatial resolution, which increases computational demands [266].

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Diffuse-interface methods: Approaches such as the Volume of Fluid (VOF) and Phase-Field Models represent interfaces over a finite thickness, allowing for more robust simulations on coarser grids but potentially introducing numerical diffusion [267].

In boiling simulations, capturing nucleation site dynamics and bubble interactions is essential. This often requires subgrid-scale models or hybrid approaches that combine empirical correlations with direct numerical simulations [268,269]. For solidification processes, accurately modeling latent heat release and anisotropic thermal conductivity is vital, processes which may require adaptive meshing and advanced solvers to balance accuracy with computational efficiency [270,271].

Traditional methodologies, exemplified by the Conjugate Gradient Method (CGM), face significant challenges in addressing transient heat-transfer phenomena, primarily because of the presence of noise and inherent instability. On the contrary, advanced deep learning techniques, including DNNs, have demonstrated considerable potential to mitigate these difficulties by producing more precise predictions, even in the presence of noisy datasets [272]. Peridynamic models offer a unique approach to transient heat transfer by using large horizon sizes to approximate local solutions. This method can effectively handle transient behaviors in homogeneous and heterogeneous materials, although it requires careful implementation of boundary conditions [273]. The Finite Pointset Method (FPM) constitutes a non-mesh-based approach that significantly improves numerical stability and computational efficiency in the analysis of three-dimensional transient heat conduction phenomena. This methodology is especially proficient in dealing with intricate geometrical configurations and transient states, as exemplified by the behaviors observed in functionally graded materials [274]. Semi-analytical approaches have been formulated to investigate transient thermal transfer phenomena in laminated constructs characterized by time-dependent boundary conditions. These methodologies are adept at accommodating various types of waveforms, including square, triangular, and sinusoidal waveforms, thus elucidating the influence of these conditions on thermal distribution profiles [275]. In data centers, the boundary conditions exhibit considerable dynamism as a result of variations in server power consumption and cooling rates. CFD models serve as a key tool in simulations of these conditions; however, they require comprehensive case studies to effectively elucidate transient temperature fluctuations [276]. Practical thermal processes often involve complex phenomena and materials with large changes in properties, making it difficult to accurately define boundary conditions. This complexity requires advanced numerical methods and careful modeling to achieve reliable simulations [2]. In civil engineering, heat-transfer problems often involve coupled modes of heat transfer, such as convection, conduction, and radiation. Methods such as the Lattice Boltzmann Method and radiosity are used to simulate these complex interactions, but require significant computational resources and expertise [277].

Although advances in numerical modeling for heat transfer have been substantial, difficulties persist in precisely representing transient phenomena and fluctuating boundary conditions. These difficulties are exacerbated by the intricate nature of actual systems and the need for advanced numerical methodologies. However, novel approaches, including deep learning and mesh-free techniques, present avenues encouraging for the addressing of these constraints, which may ultimately result in models that are more accurate and reliable in the future.

Numerical modeling for heat transfer faces significant challenges, particularly in the context of data availability and quality for machine learning (ML) applications. These challenges stem from the complexity of thermal systems, the need for high-quality data, and the limitations of current ML models in handling such data. The integration of ML in thermal modeling offers potential solutions but also highlights several obstacles that need to be addressed to improve the accuracy and reliability of the model.

High-quality data are often scarce in heat-transfer applications, which complicates the training of ML models. This scarcity is exacerbated by inconsistencies between published data sources and the complex influence of inputs, which are often correlated [278]. In real-world applications, thermal boundary conditions are frequently imprecise, leading to poorly defined conditions for energy equations. This lack of accurate data makes it difficult to solve heat-transfer problems effectively [279]. ML models, while powerful, have limited extrapolation capabilities to unseen conditions. This limitation is a significant challenge when deploying these models in heat-transfer applications, as they may not be generalized much beyond training data [278]. In the context of nuclear applications, ANNs have shown a remarkable capacity to substantially decrease computational duration compared with conventional numerical simulations, thus underscoring their potential utility for real-time thermal optimization [199].

Although machine learning presents prospective methodologies to improve thermal modeling, the obstacles associated with data quality and availability continue to constitute considerable impediments. The resolution of these challenges requires a methodical strategy for data acquisition and model formulation, thus guaranteeing that machine learning models exhibit both precision and generalizability under various conditions.

Numerical modeling for heat transfer presents significant challenges, particularly in balancing computational demands with the need for sustainable simulations. These challenges arise from the complexity of heat-transfer processes, which often involve conduction, convection, and radiation, each requiring different modeling approaches and computational resources. The development of accurate yet computationally efficient models is crucial for applications ranging from industrial processes to environmental studies. This response explores the key challenges and strategies in numerical modeling for heat transfer, focusing on computational demands and sustainability.

Radiative heat transfer, especially in media with absorbing and scattering properties, is computationally intensive. Calculations must be performed across a wide spectral range, and the integrodifferential radiative transfer equation (RTE) is complex to solve. Simplified models, such as the transport approximation, can reduce computational demands while maintaining precision in certain scenarios [259]. Convection presents considerable difficulties in modeling, owing to its intrinsic reliance on fluid dynamics. The application of finite element analysis (FEA) and various numerical methodologies requires substantial computational resources and specialized knowledge. Streamlined models, which replace convection mechanisms with thermal sources and sinks, can yield precise results while minimizing computational demands [280]. Validating numerical models with experimental data is crucial for ensuring accuracy. For example, in urban heat modeling, the use of distributed meteorological sensor networks can enhance model validation, improving the reliability of simulations [281]. Techniques like dynamic mesh optimization (DMO) can significantly reduce computational costs. In aquifer thermal energy storage (ATES) systems, DMO has been shown to reduce the number of mesh elements by up to 22 times and the simulation time by up to 15 times, without sacrificing precision [282]. In turbulent flow simulations, such as those involving eccentric co-rotating heat transfer, incorporating new source terms in turbulence models can enhance accuracy. These improvements help to capture complex flow dynamics and temperature distributions more effectively [283]. A laser beam heating model (LBHM) was developed for the platform for numerical modeling associated with a multi-material selective laser melting process. The LBHM is utilized as a ray-tracing algorithm. The model was developed for transparent and translucent materials, taking into account phenomena such as refraction, scattering, and volume absorption [284].

Although numerical modeling of heat transfer encounters considerable obstacles, advances in computational methodologies and model simplifications present promising avenues for resolution. However, it is crucial to acknowledge that these models frequently involve compromises between precision and computational efficiency. Furthermore, the intrinsic biases present in modeling software and the need for high-fidelity simulations emphasize the critical need for the ongoing validation and enhancement of numerical models. As computational capabilities expand and novel techniques emerge, the prospects for more sustainable and precise heat-transfer simulations are anticipated to grow, thereby benefiting a diverse array of applications ranging from industrial operations to environmental management.

7. Future Directions and Emerging Trends

The future of numerical heat-transfer modeling is shaped by several emerging trends, including quantum computing, real-time adaptive models, green computing, and open-source collaborative platforms. To enhance clarity and coherence, these trends are structured into distinct sections, each exploring key advancements, methodologies, and challenges.

7.1. Quantum Computing for Thermal Simulations

Quantum computing holds significant promise for overcoming computational challenges in thermal simulations, offering breakthroughs in efficiency and accuracy. The application of quantum algorithms to thermal modeling has the potential to revolutionize the simulation of heat conduction, thermal field theories, and many-body systems, models which are computationally demanding on classical computers.

One of the key advantages of quantum computing is its ability to solve complex differential equations exponentially faster than classical methods. Quantum algorithms, such as the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA), have been explored for their potential to model complex differential equations with exponential speedup over classical methods [285]. Specifically, the application of quantum Fourier transforms and Hamiltonian simulation techniques has demonstrated efficiency in solving high-dimensional PDEs, including the heat equation and the Schrödinger equation [286,287]. These approaches leverage quantum parallelism and entanglement to reduce computational complexity, making them promising for large-scale simulations.

In another example, quantum algorithms can efficiently solve the one-dimensional heat equation with Dirichlet boundary conditions by encoding initial temperature distributions into quantum states and simulating their evolution using quantum gates. This approach uses the Trotter–Suzuki decomposition to model heat propagation, achieving exponential computational speedup as the counts of qubit increase [288]. Studies indicate that quantum simulations can reduce relative errors in heat conduction models, particularly when using quantum statevector simulators, which outperform other quantum devices in accuracy.

Beyond heat conduction, quantum computing is also being explored in fermionic field theories, in which quantum imaginary time evolution algorithms enable the evaluation of thermal distributions and energy densities, closely aligning with analytical and semi-classical expectations [289]. These quantum simulations provide new insights into thermal fixed points and are expected to advance real-time thermalization studies, which are crucial in strongly interacting systems at finite temperatures.

Quantum computing is also revolutionizing the study of condensed matter systems, such as the Fermi–Hubbard model, by employing variational quantum algorithms to optimize circuit designs and mitigate coherence limitations. These techniques allow for the simulation of high-temperature materials, even though current quantum hardware still faces challenges related to error sources and barren plateaus [290].

A key development in quantum thermal simulations is the ability to study thermalization dynamics in lattice gauge theories, particularly in relation to quantum chaos and entanglement. Recent research has demonstrated that randomized measurement protocols enable the efficient learning of classical approximations of non-equilibrium states, revealing universal thermalization properties [291]. This solidifies quantum computing as a valuable tool for analyzing non-equilibrium processes in complex many-body systems.

Beyond these advantages, quantum algorithms can also facilitate matrix computations, particularly in solving large linear systems using the Harrow–Hassidim–Lloyd (HHL) algorithm. This algorithm offers exponential speedup in solving well-conditioned sparse linear systems, which are common in numerical modeling of PDEs [292]. The integration of quantum solvers with classical preconditioning methods is a promising direction for the enhancement of numerical stability and performance. Recent studies highlight that hybrid quantum–classical algorithms, such as Quantum Singular Value Transformation (QSVT), further improve the feasibility of quantum-enhanced linear solvers by mitigating noise and reducing algorithmic complexity [293].

Furthermore, quantum computing holds promise for spectral methods and eigenvalue problems, which are critical in numerical modeling. The Quantum Phase Estimation (QPE) algorithm enables efficient computation of eigenvalues of large matrices, which provides advantages in solving high-dimensional PDE and fluid dynamics problems [294]. By utilizing quantum superposition and interference, QPE allows for more precise eigenvalue estimation compared to classical iterative methods [292]. This is particularly relevant in turbulence modeling and lattice gauge theory simulations, in which eigenvalue calculations are computationally demanding [295].

In addition, hybrid quantum–classical algorithms are being employed to simulate thermal statistical states, such as Gibbs states, using superconducting quantum processors. These methods combine classical probability models with variational quantum circuits, allowing researchers to prepare excited states and estimate thermal observables with high fidelity [296]. The scalability of these approaches, along with their self-verifying features, indicates a strong potential for solving large-scale quantum statistical mechanics problems.

Despite its immense potential, quantum computing in thermal simulations still faces significant challenges, particularly in terms of hardware limitations, coherence constraints, and error mitigation strategies. Quantum systems suffer from decoherence, and require robust fault-tolerant algorithms to ensure accurate results. Additionally, the practical implementation of large-scale quantum thermal simulations is constrained by current quantum processor capabilities.

Recent research suggests that the quantum advantage is most evident in linear algebraic problems, such as solving large sparse matrices and eigenvalue calculations, but remains limited for highly nonlinear systems due to the overhead associated with quantum error correction and qubit entanglement. As quantum hardware continues to evolve, improvements in qubit connectivity and fault-tolerant architectures are expected to further extend the applicability of quantum computing to numerical modeling.

7.2. Real-Time Adaptive Models

Real-time adaptive models are transforming numerical heat-transfer simulations by enabling rapid adjustments to dynamic systems. These models employ advanced computational techniques to respond to fluctuating conditions instantaneously, significantly improving simulation speed and precision. The integration of machine learning (ML) and adaptive algorithms plays a crucial role in enhancing the functionality and efficiency of these models.

One notable advancement is the Adaptive Universal Physics-Guided Auto-Solver (AUPgAS), a major development within the field of Physics-Informed Neural Networks (PINNs). AUPgAS incorporates operational conditions as variables within the neural network, increasing flexibility while dramatically reducing training time. Studies have shown that AUPgAS can simulate homogeneous heat flow problems in 3.4 s, compared to 910 s with traditional methods, while maintaining a minimum average error of 13.55% [297].

Another innovative approach is the ANN-VAE composite model, which enables highly efficient three-dimensional heat and mass transfer predictions. This model improves computational efficiency by accelerating prediction speeds by nearly 380,000 times over conventional techniques, while achieving a mean accuracy of 97.3% for temperature predictions and 97.9% for velocity field estimates [298].

To address the high computational cost of multiphysics simulations, the Local Moving Thermal–Fluid Framework has been developed. This method focuses computational efforts on a localized moving zone containing the melt pool, while solving the heat-transfer problem across the entire domain. It has shown high fidelity, maintaining a relative error of less than 2.6%, while reducing the simulation time from weeks to just 62 h in large-scale simulations [299].

Another key development is Anisotropic Mesh Adaptation, which enhances the accuracy of finite element simulations for radiative heat transfer. This approach employs hierarchical error estimators and long time steps to effectively resolve internal boundary layers in anisotropic media, leading to improved computational efficiency without compromising precision [300].

Despite the significant advancements in adaptive real-time models, several challenges persist, particularly in balancing computational efficiency with model accuracy. The integration of emerging technologies, such as quantum computing and cloud-based resources, presents opportunities to further enhance these models by enabling faster and more scalable simulations. Furthermore, the trade-offs between spatial and temporal resolution, as resolution, as observed in adaptive imaging systems, highlight the need for the continued refinement and validation of these models to ensure their robustness and applicability across diverse engineering contexts.

7.3. Green Computing and Sustainable Numerical Modeling

Future directions in numerical heat-transfer modeling are increasingly intertwined with green computing initiatives aiming to enhance computational efficiency and reduce environmental impact. This integration is crucial as the demand for more sophisticated and accurate heat-transfer models grows, particularly in applications such as energy-efficient building design, advanced heat exchangers, and urban climate modeling. Green computing principles can be applied to optimize these models, to ensure that they are both effective and sustainable.

Machine learning methodologies, including extreme gradient boost and support vector regression, have demonstrated considerable potential in enhancing the efficiency and precision of thermal transfer models. These approaches can markedly reduce the computational duration and energy expenditure compared to conventional numerical simulations, aligning with the objectives of sustainable computing [123].

The development of green nanofluids, such as CGNPs/H₂O, for CPU cooling systems exemplifies the application of sustainable materials in heat transfer. These nanofluids improve thermal efficiency while minimizing environmental impact, demonstrating a practical application of the principles of green computing [301]. Additionally, the integration of phase change materials (PCMs) in thermal energy storage systems for air conditioning can significantly reduce energy consumption and emissions. These systems take advantage of the high thermal conductivity of materials such as expanded graphite to optimize heat transfer, supporting the goals of sustainable development [302].

In addition, numerical models of heat pipes are being optimized to enhance heat exchange processes while reducing the size and energy requirements of heat exchangers. This aligns with the objectives of green computing by promoting energy efficiency and reducing material usage [303]. Another area of progress is the incorporation of moisture transfer into urban climate models, which can improve the precision of energy consumption predictions and indoor climate simulations. This approach requires efficient numerical methods to balance computational cost and accuracy, a key consideration in green computing [304].

Although the incorporation of sustainable computing practices into quantitative heat-transfer modeling presents numerous advantages, several obstacles persist. The intricacy of precisely simulating heat-transfer phenomena, particularly within heterogeneous and dynamic contexts, can result in higher computational requirements. Achieving a balance between the necessity for comprehensive simulations and the tenet of sustainable computing requires continuous academic inquiry and technological advancement. Furthermore, the implementation of novel materials and methodologies must be rigorously assessed to ascertain whether they do not inadvertently engender unforeseen ecological repercussions.

7.4. Open-Source and Collaborative Platforms

The future of numerical heat-transfer modeling will be significantly shaped by the advent of open-source tools and collaborative platforms that are democratizing access to advanced computational methods. These resources enable researchers and engineers to tackle complex thermal management challenges more efficiently and cost-effectively. Open-source software (OSS) and collaborative platforms foster innovation by providing flexible, scalable, and accessible solutions for modeling heat-transfer phenomena. This transition is particularly impactful in fields such as electronics cooling, vehicle thermal management, and data center optimization.

One of the most significant advantages of open-source platforms is the elimination of financial barriers associated with commercial software licenses, making advanced modeling techniques accessible to a broader audience. For example, the integration of Firedrake, PETSc, GMSH, and Paraview in the thermal analysis of finite elements enables high-fidelity simulations of complex heat sinks without the need for expensive proprietary software [37]. This accessibility improves collaboration and innovation, as researchers can build on existing frameworks rather than developing solutions from scratch.

Open-source platforms also offer flexibility in model customization, which is crucial for adapting simulations to new cooling solutions, unconventional heat-transfer mechanisms, and complex geometries. This adaptability is demonstrated in the development of open-hardware platforms for thermal modeling, which provide a means to validate and refine models under realistic conditions [305]. These platforms facilitate cross-disciplinary collaboration and accelerate the development of novel thermal management strategies.

Furthermore, the integration of data-driven and equation-based modeling tools within open-source frameworks is gaining momentum. These hybrid approaches significantly improve predictive capabilities in thermal management systems, allowing for better accuracy and reduced computational costs [306]. For example, graph neural networks (GNNs) have been applied in the design of thermal management systems, streamlining the optimization process. Using GNNs, researchers can quickly identify optimal design candidates, significantly reducing the computational burden associated with exhaustive search methods [307].

In addition to modeling advancements, open-source initiatives have led to the development of Fast Fluid Dynamics (FFD) models, which offer an efficient alternative to traditional CFD simulations for thermal management in data centers. These models provide comparable precision with significantly reduced computational time, making them a promising tool for efficient data center cooling and thermal regulation strategies [308].

Despite these advantages, the long-term maintenance and sustainability of open-source projects remain key challenges. Ensuring continued development, user support, and software updates is critical to sustain the momentum of open-source platforms. In addition, integrating open-source tools into existing industrial workflows requires careful consideration of compatibility, user training, and validation against established methodologies.

As these challenges are addressed, the roles of open-source and collaborative platforms in advancing numerical heat-transfer modeling are expected to expand. By fostering an ecosystem of shared knowledge, reusable tools, and collaborative innovation, these platforms will play an essential role in shaping the future of thermal management research and engineering.

8. Conclusions

The review comprehensively highlights significant advancements in numerical modeling techniques for heat transfer, encompassing methods like Computational Fluid Dynamics (CFD), Finite Element Methods (FEM), and Finite Volume Methods (FVM), as well as emerging strategies such as Adaptive Mesh Refinement (AMR), machine learning (ML), and high-performance computing (HPC). These developments have elevated the accuracy, efficiency, and applicability of thermal simulations in various domains, including electronics, renewable energy, and sustainable construction. A key contribution of this study lies in its detailed exploration of the interplay between these techniques, particularly their integration with multiphysics modeling and reduced-order approaches, which significantly improve computational efficiency in solving complex thermal problems.

A critical theme throughout the review is the environmental impact of these computational advances, particularly the energy consumption and carbon footprint associated with large-scale simulations. The review underscores the importance of sustainable computing practices, such as efficient algorithms, energy-efficient hardware, and the use of renewable energy in data centers, to mitigate these effects. By emphasizing these sustainability concerns, this study presents innovative perspectives on how computational modeling can align with global environmental goals. These strategies are aligned with the growing need to balance computational demands with environmental sustainability.

Furthermore, this study highlights the importance of interdisciplinary collaboration in overcoming challenges related to computational cost, model accuracy, and the integration of real-world complexities in thermal simulations. Looking ahead, the review calls for intensive interdisciplinary collaboration and research to overcome persistent challenges in the field, including balancing computational cost with accuracy, addressing real-world complexities, and improving data quality for AI-driven models. The exploration of quantum computing, real-time adaptive modeling, and green computing initiatives signals promising pathways for future advancements. Furthermore, this work underscores the role of open-source and collaborative platforms as vital enablers for democratizing access to advanced computational tools and fostering innovation in heat-transfer modeling. This enhanced perspective provides a structured roadmap to advance the field while emphasizing the imperative of sustainable practices.